Fundamental theorem of algebra

 

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The fundamental theorem of algebra, also known as d'Alembert's theorem, or the d'Alembert–Gauss theorem, states that every non-constant single-variable polynomial with complex coefficients has at least one complex root. This includes polynomials with real coefficients, since every real number is a complex number with its imaginary part equal to zero.

Equivalently (by definition), the theorem states that the field of complex numbers is algebraically closed.

The theorem is also stated as follows: every non-zero, single-variable, degree n polynomial with complex coefficients has, counted with multiplicity, exactly n complex roots. The equivalence of the two statements can be proven through the use of successive polynomial division.

Despite its name, there is no purely algebraic proof of the theorem, since any proof must use some form of the analytic completeness of the real numbers, which is not an algebraic concept. Additionally, it is not fundamental for modern algebra; its name was given at a time when algebra was synonymous with theory of equations.

History

Peter Roth, in his book Arithmetica Philosophica (published in 1608, at Nürnberg, by Johann Lantzenberger), wrote that a polynomial equation of degree n (with real coefficients) may have n solutions. Albert Girard, in his book L'invention nouvelle en l'Algèbre (published in 1629), asserted that a polynomial equation of degree n has n solutions, but he did not state that they had to be real numbers. Furthermore, he added that his assertion holds "unless the equation is incomplete", by which he meant that no coefficient is equal to 0. However, when he explains in detail what he means, it is clear that he actually believes that his assertion is always true; for instance, he shows that the equation ${\displaystyle x^4=4x-3,}$ although incomplete, has four solutions (counting multiplicities): 1 (twice), ${\displaystyle -1+i\sqrt{2},}$ and ${\displaystyle -1-i\sqrt{2}.}$

As will be mentioned again below, it follows from the fundamental theorem of algebra that every non-constant polynomial with real coefficients can be written as a product of polynomials with real coefficients whose degrees are either 1 or 2. However, in 1702 Leibniz erroneously said that no polynomial of the type x⁴ + a⁴ (with a real and distinct from 0) can be written in such a way. Later, Nikolaus Bernoulli made the same assertion concerning the polynomial x⁴ − 4x³ + 2x² + 4x + 4, but he got a letter from Euler in 1742 in which it was shown that this polynomial is equal to

${\displaystyle \left(x^2-(2+\alpha )x+1+\sqrt{7}+\alpha \right)\left(x^2-(2-\alpha )x+1+\sqrt{7}-\alpha \right),}$

with ${\displaystyle \alpha =\sqrt{4+2\sqrt{7}}.}$ Also, Euler pointed out that

${\displaystyle x^4+a^4=\left(x^2+a\sqrt{2}\cdot x+a^2\right)\left(x^2-a\sqrt{2}\cdot x+a^2\right).}$

A first attempt at proving the theorem was made by d'Alembert in 1746, but his proof was incomplete. Among other problems, it assumed implicitly a theorem (now known as Puiseux's theorem), which would not be proved until more than a century later and using the fundamental theorem of algebra. Other attempts were made by Euler (1749), de Foncenex (1759), Lagrange (1772), and Laplace (1795). These last four attempts assumed implicitly Girard's assertion; to be more precise, the existence of solutions was assumed and all that remained to be proved was that their form was a + bi for some real numbers a and b. In modern terms, Euler, de Foncenex, Lagrange, and Laplace were assuming the existence of a splitting field of the polynomial p(z).

At the end of the 18th century, two new proofs were published which did not assume the existence of roots, but neither of which was complete. One of them, due to James Wood and mainly algebraic, was published in 1798 and it was totally ignored. Wood's proof had an algebraic gap. The other one was published by Gauss in 1799 and it was mainly geometric, but it had a topological gap, only filled by Alexander Ostrowski in 1920, as discussed in Smale (1981).

The first rigorous proof was published by Argand, an amateur mathematician, in 1806 (and revisited in 1813); it was also here that, for the first time, the fundamental theorem of algebra was stated for polynomials with complex coefficients, rather than just real coefficients. Gauss produced two other proofs in 1816 and another incomplete version of his original proof in 1849.

The first textbook containing a proof of the theorem was Cauchy's Cours d'analyse de l'École Royale Polytechnique (1821). It contained Argand's proof, although Argand is not credited for it.

None of the proofs mentioned so far is constructive. It was Weierstrass who raised for the first time, in the middle of the 19th century, the problem of finding a constructive proof of the fundamental theorem of algebra. He presented his solution, which amounts in modern terms to a combination of the Durand–Kerner method with the homotopy continuation principle, in 1891. Another proof of this kind was obtained by Hellmuth Kneser in 1940 and simplified by his son Martin Kneser in 1981.

Without using countable choice, it is not possible to constructively prove the fundamental theorem of algebra for complex numbers based on the Dedekind real numbers (which are not constructively equivalent to the Cauchy real numbers without countable choice). However, Fred Richman proved a reformulated version of the theorem that does work.

Equivalent statements

There are several equivalent formulations of the theorem:

• Every univariate polynomial of positive degree with real coefficients has at least one complex root.
• Every univariate polynomial of positive degree with complex coefficients has at least one complex root.

  This implies immediately the previous assertion, as real numbers are also complex numbers. The converse results from the fact that one gets a polynomial with real coefficients by taking the product of a polynomial and its complex conjugate (obtained by replacing each coefficient with its complex conjugate). A root of this product is either a root of the given polynomial, or of its conjugate; in the latter case, the conjugate of this root is a root of the given polynomial.

• Every univariate polynomial of positive degree n with complex coefficients can be factorized as
  $${\displaystyle c(x-r_1)\cdots (x-r_n),}$$

  
  where ${\displaystyle c,r_1,\dots ,r_n}$ are complex numbers.

  The n complex numbers ${\displaystyle r_1,\dots ,r_n}$ are the roots of the polynomial. If a root appears in several factors, it is a multiple root, and the number of its occurrences is, by definition, the multiplicity of the root.

  The proof that this statement results from the previous ones is done by recursion on n: when a root ${\displaystyle r_1}$ has been found, the polynomial division by ${\displaystyle x-r_1}$ provides a polynomial of degree ${\displaystyle n-1}$ whose roots are the other roots of the given polynomial.

The next two statements are equivalent to the previous ones, although they do not involve any nonreal complex number. These statements can be proved from previous factorizations by remarking that, if r is a non-real root of a polynomial with real coefficients, its complex conjugate ${\displaystyle \overline{r}}$ is also a root, and ${\displaystyle (x-r)(x-\overline{r})}$ is a polynomial of degree two with real coefficients (this is the complex conjugate root theorem). Conversely, if one has a factor of degree two, the quadratic formula gives a root.

• Every univariate polynomial with real coefficients of degree larger than two has a factor of degree two with real coefficients.
• Every univariate polynomial with real coefficients of positive degree can be factored as
  $${\displaystyle c{p}_1\cdots p_k,}$$

  
  where c is a real number and each ${\displaystyle p_i}$ is a monic polynomial of degree at most two with real coefficients. Moreover, one can suppose that the factors of degree two do not have any real root.

Proofs

All proofs below involve some mathematical analysis, or at least the topological concept of continuity of real or complex functions. Some also use differentiable or even analytic functions. This requirement has led to the remark that the Fundamental Theorem of Algebra is neither fundamental, nor a theorem of algebra.

Some proofs of the theorem only prove that any non-constant polynomial with real coefficients has some complex root. This lemma is enough to establish the general case because, given a non-constant polynomial p with complex coefficients, the polynomial

${\displaystyle q=p\overline{p},}$

has only real coefficients, and, if z is a root of q, then either z or its conjugate is a root of p. Here, ${\displaystyle \overline{p}}$ is the polynomial obtaining by replacing each coefficient of p with its complex conjugate; the roots of ${\displaystyle \overline{p}}$ are exactly the complex conjugates of the roots of p

Many non-algebraic proofs of the theorem use the fact (sometimes called the "growth lemma") that a polynomial function p(z) of degree n whose dominant coefficient is 1 behaves like zⁿ when |z| is large enough. More precisely, there is some positive real number R such that

${\displaystyle \frac{1}{2}|z^n|<|p(z)|<\frac{3}{2}|z^n|}$

when |z| > R.

Real-analytic proofs

Even without using complex numbers, it is possible to show that a real-valued polynomial p(x): p(0) ≠ 0 of degree n > 2 can always be divided by some quadratic polynomial with real coefficients. In other words, for some real-valued a and b, the coefficients of the linear remainder on dividing p(x) by x² − ax − b simultaneously become zero.

${\displaystyle p(x)=(x^2-ax-b)q(x)+x{R}_{p(x)}(a,b)+S_{p(x)}(a,b),}$

where q(x) is a polynomial of degree n − 2. The coefficients R_p(x)(a, b) and S_p(x)(a, b) are independent of x and completely defined by the coefficients of p(x). In terms of representation, R_p(x)(a, b) and S_p(x)(a, b) are bivariate polynomials in a and b. In the flavor of Gauss's first (incomplete) proof of this theorem from 1799, the key is to show that for any sufficiently large negative value of b, all the roots of both R_p(x)(a, b) and S_p(x)(a, b) in the variable a are real-valued and alternating each other (interlacing property). Utilizing a Sturm-like chain that contain R_p(x)(a, b) and S_p(x)(a, b) as consecutive terms, interlacing in the variable a can be shown for all consecutive pairs in the chain whenever b has sufficiently large negative value. As S_p(a, b = 0) = p(0) has no roots, interlacing of R_p(x)(a, b) and S_p(x)(a, b) in the variable a fails at b = 0. Topological arguments can be applied on the interlacing property to show that the locus of the roots of R_p(x)(a, b) and S_p(x)(a, b) must intersect for some real-valued a and b < 0.

Complex-analytic proofs

Find a closed disk D of radius r centered at the origin such that |p(z)| > |p(0)| whenever |z| ≥ r. The minimum of |p(z)| on D, which must exist since D is compact, is therefore achieved at some point z₀ in the interior of D, but not at any point of its boundary. The maximum modulus principle applied to 1/p(z) implies that p(z₀) = 0. In other words, z₀ is a zero of p(z).

A variation of this proof does not require the maximum modulus principle (in fact, a similar argument also gives a proof of the maximum modulus principle for holomorphic functions). Continuing from before the principle was invoked, if a := p(z₀) ≠ 0, then, expanding p(z) in powers of z − z₀, we can write

${\displaystyle p(z)=a+c_k(z-z_0{)}^k+c_{k+1}(z-z_0{)}^{k+1}+\cdots +c_n(z-z_0{)}^n.}$

Here, the c_j are simply the coefficients of the polynomial z → p(z + z₀) after expansion, and k is the index of the first non-zero coefficient following the constant term. For z sufficiently close to z₀ this function has behavior asymptotically similar to the simpler polynomial ${\displaystyle q(z)=a+c_k(z-z_0{)}^k}$. More precisely, the function

${\displaystyle \left|\frac{p(z)-q(z)}{(z-z_0{)}^{k+1}}\right|\le M}$

for some positive constant M in some neighborhood of z₀. Therefore, if we define ${\displaystyle {\theta }_0=(\arg (a)+\pi -\arg (c_k))/k}$ and let ${\displaystyle z=z_0+r{e}^{i{\theta }_0}}$ tracing a circle of radius r > 0 around z, then for any sufficiently small r (so that the bound M holds), we see that

${\displaystyle \begin{array}{rl}|p(z)|& \le |q(z)|+r^{k+1}\left|\frac{p(z)-q(z)}{r^{k+1}}\right|\\ & \le \left|a+(-1)c_k{r}^k{e}^{i(\arg (a)-\arg (c_k))}\right|+M{r}^{k+1}\\ & =|a|-|c_k|r^k+M{r}^{k+1}\end{array}}$

When r is sufficiently close to 0 this upper bound for |p(z)| is strictly smaller than |a|, contradicting the definition of z₀. Geometrically, we have found an explicit direction θ₀ such that if one approaches z₀ from that direction one can obtain values p(z) smaller in absolute value than |p(z₀)|.

Another analytic proof can be obtained along this line of thought observing that, since |p(z)| > |p(0)| outside D, the minimum of |p(z)| on the whole complex plane is achieved at z₀. If |p(z₀)| > 0, then 1/p is a bounded holomorphic function in the entire complex plane since, for each complex number z, |1/p(z)| ≤ |1/p(z₀)|. Applying Liouville's theorem, which states that a bounded entire function must be constant, this would imply that 1/p is constant and therefore that p is constant. This gives a contradiction, and hence p(z₀) = 0.

Yet another analytic proof uses the argument principle. Let R be a positive real number large enough so that every root of p(z) has absolute value smaller than R; such a number must exist because every non-constant polynomial function of degree n has at most n zeros. For each r > R, consider the number

${\displaystyle \frac{1}{2\pi i}{\int }_{c(r)}\frac{p^{\prime }(z)}{p(z)}dz,}$

where c(r) is the circle centered at 0 with radius r oriented counterclockwise; then the argument principle says that this number is the number N of zeros of p(z) in the open ball centered at 0 with radius r, which, since r > R, is the total number of zeros of p(z). On the other hand, the integral of n/z along c(r) divided by 2πi is equal to n. But the difference between the two numbers is

${\displaystyle \frac{1}{2\pi i}{\int }_{c(r)}\left(\frac{p^{\prime }(z)}{p(z)}-\frac{n}{z}\right)dz=\frac{1}{2\pi i}{\int }_{c(r)}\frac{z{p}^{\prime }(z)-np(z)}{zp(z)}dz.}$

The numerator of the rational expression being integrated has degree at most n − 1 and the degree of the denominator is n + 1. Therefore, the number above tends to 0 as r → +∞. But the number is also equal to N − n and so N = n.

Another complex-analytic proof can be given by combining linear algebra with the Cauchy theorem. To establish that every complex polynomial of degree n > 0 has a zero, it suffices to show that every complex square matrix of size n > 0 has a (complex) eigenvalue. The proof of the latter statement is by contradiction.

Let A be a complex square matrix of size n > 0 and let I_n be the unit matrix of the same size. Assume A has no eigenvalues. Consider the resolvent function

${\displaystyle R(z)=(z{I}_n-A{)}^{-1},}$

which is a meromorphic function on the complex plane with values in the vector space of matrices. The eigenvalues of A are precisely the poles of R(z). Since, by assumption, A has no eigenvalues, the function R(z) is an entire function and Cauchy theorem implies that

${\displaystyle {\int }_{c(r)}R(z)dz=0.}$

On the other hand, R(z) expanded as a geometric series gives:

${\displaystyle R(z)=z^{-1}(I_n-z^{-1}A{)}^{-1}=z^{-1}\sum _{k=0}^{\mathrm{\infty }}\frac{1}{z^k}{A}^k\cdot }$

This formula is valid outside the closed disc of radius ${\displaystyle \Vert A\Vert }$ (the operator norm of A). Let ${\displaystyle r>\Vert A\Vert .}$ Then

${\displaystyle {\int }_{c(r)}R(z)dz=\sum _{k=0}^{\mathrm{\infty }}{\int }_{c(r)}\frac{dz}{z^{k+1}}{A}^k=2\pi i{I}_n}$

(in which only the summand k = 0 has a nonzero integral). This is a contradiction, and so A has an eigenvalue.

Finally, Rouché's theorem gives perhaps the shortest proof of the theorem.

Topological proofs

Animation illustrating the proof on the polynomial ${\displaystyle x^5-x-1}$

Suppose the minimum of |p(z)| on the whole complex plane is achieved at z₀; it was seen at the proof which uses Liouville's theorem that such a number must exist. We can write p(z) as a polynomial in z − z₀: there is some natural number k and there are some complex numbers c_k, c_k + 1, ..., c_n such that c_k ≠ 0 and:

${\displaystyle p(z)=p(z_0)+c_k(z-z_0{)}^k+c_{k+1}(z-z_0{)}^{k+1}+\cdots +c_n(z-z_0{)}^n.}$

If p(z₀) is nonzero, it follows that if a is a k^th root of −p(z₀)/c_k and if t is positive and sufficiently small, then |p(z₀ + ta)| < |p(z₀)|, which is impossible, since |p(z₀)| is the minimum of |p| on D.

For another topological proof by contradiction, suppose that the polynomial p(z) has no roots, and consequently is never equal to 0. Think of the polynomial as a map from the complex plane into the complex plane. It maps any circle |z| = R into a closed loop, a curve P(R). We will consider what happens to the winding number of P(R) at the extremes when R is very large and when R = 0. When R is a sufficiently large number, then the leading term zⁿ of p(z) dominates all other terms combined; in other words,

${\displaystyle \left|z^n\right|>\left|a_{n-1}{z}^{n-1}+\cdots +a_0\right|.}$

When z traverses the circle ${\displaystyle R{e}^{i\theta }}$ once counter-clockwise ${\displaystyle (0\le \theta \le 2\pi ),}$ then ${\displaystyle z^n=R^n{e}^{in\theta }}$ winds n times counter-clockwise ${\displaystyle (0\le \theta \le 2\pi n)}$ around the origin (0,0), and P(R) likewise. At the other extreme, with |z| = 0, the curve P(0) is merely the single point p(0), which must be nonzero because p(z) is never zero. Thus p(0) must be distinct from the origin (0,0), which denotes 0 in the complex plane. The winding number of P(0) around the origin (0,0) is thus 0. Now changing R continuously will deform the loop continuously. At some R the winding number must change. But that can only happen if the curve P(R) includes the origin (0,0) for some R. But then for some z on that circle |z| = R we have p(z) = 0, contradicting our original assumption. Therefore, p(z) has at least one zero.

Algebraic proofs

These proofs of the Fundamental Theorem of Algebra must make use of the following two facts about real numbers that are not algebraic but require only a small amount of analysis (more precisely, the intermediate value theorem in both cases):

• every polynomial with an odd degree and real coefficients has some real root;
• every non-negative real number has a square root.

The second fact, together with the quadratic formula, implies the theorem for real quadratic polynomials. In other words, algebraic proofs of the fundamental theorem actually show that if R is any real-closed field, then its extension C = R(√−1) is algebraically closed.

By induction

As mentioned above, it suffices to check the statement "every non-constant polynomial p(z) with real coefficients has a complex root". This statement can be proved by induction on the greatest non-negative integer k such that 2^k divides the degree n of p(z). Let a be the coefficient of zⁿ in p(z) and let F be a splitting field of p(z) over C; in other words, the field F contains C and there are elements z₁, z₂, ..., z_n in F such that

${\displaystyle p(z)=a(z-z_1)(z-z_2)\cdots (z-z_n).}$

If k = 0, then n is odd, and therefore p(z) has a real root. Now, suppose that n = 2^km (with m odd and k > 0) and that the theorem is already proved when the degree of the polynomial has the form 2^k − 1m′ with m′ odd. For a real number t, define:

${\displaystyle q_t(z)=\prod _{1\le i<j\le n}\left(z-z_i-z_j-t{z}_i{z}_j\right).}$

Then the coefficients of q_t(z) are symmetric polynomials in the z_i with real coefficients. Therefore, they can be expressed as polynomials with real coefficients in the elementary symmetric polynomials, that is, in −a₁, a₂, ..., (−1)ⁿa_n. So q_t(z) has in fact real coefficients. Furthermore, the degree of q_t(z) is n(n − 1)/2 = 2^k−1m(n − 1), and m(n − 1) is an odd number. So, using the induction hypothesis, q_t has at least one complex root; in other words, z_i + z_j + tz_iz_j is complex for two distinct elements i and j from {1, ..., n}. Since there are more real numbers than pairs (i, j), one can find distinct real numbers t and s such that z_i + z_j + tz_iz_j and z_i + z_j + sz_iz_j are complex (for the same i and j). So, both z_i + z_j and z_iz_j are complex numbers. It is easy to check that every complex number has a complex square root, thus every complex polynomial of degree 2 has a complex root by the quadratic formula. It follows that z_i and z_j are complex numbers, since they are roots of the quadratic polynomial z² −  (z_i + z_j)z + z_iz_j.

Joseph Shipman showed in 2007 that the assumption that odd degree polynomials have roots is stronger than necessary; any field in which polynomials of prime degree have roots is algebraically closed (so "odd" can be replaced by "odd prime" and this holds for fields of all characteristics). For axiomatization of algebraically closed fields, this is the best possible, as there are counterexamples if a single prime is excluded. However, these counterexamples rely on −1 having a square root. If we take a field where −1 has no square root, and every polynomial of degree n ∈ I has a root, where I is any fixed infinite set of odd numbers, then every polynomial f(x) of odd degree has a root (since (x² + 1)^kf(x) has a root, where k is chosen so that deg(f) + 2k ∈ I). Mohsen Aliabadi generalized Shipman's result in 2013, providing an independent proof that a sufficient condition for an arbitrary field (of any characteristic) to be algebraically closed is that it has a root for every polynomial of prime degree. Theorem(Mohsen Aliabadi): Let K be a field, and suppose every prime degree polynomial in K[x] has a root in K. Then K is algebraically closed.

First, we demonstrate that the degree of any irreducible polynomial in K[x] is divisible by some prime p. If this is not the case, for the prime factors p1, ..., pn of the degree of f(x), there would exist an irreducible polynomial gi(x) not divisible by pi. In that case, consider a polynomial F(x) = f^k0(x)g^k1_1(x)...g^kn_n(x), where k0, k1, ..., kn are non-negative integers. The degree of F(x) is k0 deg f + k1 deg g1 + ... + kn deg gn, and their greatest common divisor is 1. By Dirichlet's theorem, we can choose ki so that the degree of F(x) is a prime number. Then, F(x) has a root in K, which is a contradiction. Hence, the degree of any irreducible polynomial in K[x] is divisible by a prime p.

Next, we show that there does not exist a p-th degree extension L of K. If such an extension exists, we can take an element α belonging to L \ K and consider its minimal polynomial f(x). Then, deg f(x) = p and f(x) has a root in K, which is also a contradiction. Hence, there does not exist a p-th degree extension L of K.

Finally, we show that there does not exist a finite Galois extension L of K. If such an extension exists, we denote [L : K] = p^r * m, where r and m are natural numbers and (m, p) = 1. According to Galois theory, there exists a subfield L' of L where [L : L'] = p^r. Thus, [L' : K] = m. If m > 1, we can take an element α belonging to L' \ K and consider its minimal polynomial f(x). Then deg f(x) divides m and f(x) is irreducible, hence p divides deg f(x). However, p does not divide m, which is a contradiction. Therefore, m = 1 and [L : K] = p^r. Again, according to Galois theory, there exists a subfield L' of L where [L : L'] = p^(r-1). Then [L' : K] = p, which contradicts what we proved earlier. Hence, there does not exist a finite Galois extension L of K.

From the above, we conclude that K is algebraically closed.

From Galois theory

Another algebraic proof of the fundamental theorem can be given using Galois theory. It suffices to show that C has no proper finite field extension. Let K/C be a finite extension. Since the normal closure of K over R still has a finite degree over C (or R), we may assume without loss of generality that K is a normal extension of R (hence it is a Galois extension, as every algebraic extension of a field of characteristic 0 is separable). Let G be the Galois group of this extension, and let H be a Sylow 2-subgroup of G, so that the order of H is a power of 2, and the index of H in G is odd. By the fundamental theorem of Galois theory, there exists a subextension L of K/R such that Gal(K/L) = H. As [L:R] = [G:H] is odd, and there are no nonlinear irreducible real polynomials of odd degree, we must have L = R, thus [K:R] and [K:C] are powers of 2. Assuming by way of contradiction that [K:C] > 1, we conclude that the 2-group Gal(K/C) contains a subgroup of index 2, so there exists a subextension M of C of degree 2. However, C has no extension of degree 2, because every quadratic complex polynomial has a complex root, as mentioned above. This shows that [K:C] = 1, and therefore K = C, which completes the proof.

Geometric proofs

There exists still another way to approach the fundamental theorem of algebra, due to J. M. Almira and A. Romero: by Riemannian geometric arguments. The main idea here is to prove that the existence of a non-constant polynomial p(z) without zeros implies the existence of a flat Riemannian metric over the sphere S². This leads to a contradiction since the sphere is not flat.

A Riemannian surface (M, g) is said to be flat if its Gaussian curvature, which we denote by K_g, is identically null. Now, the Gauss–Bonnet theorem, when applied to the sphere S², claims that

${\displaystyle {\int }_{{\mathbf{S}}^2}{K}_g=4\pi ,}$

which proves that the sphere is not flat.

Let us now assume that n > 0 and

${\displaystyle p(z)=a_0+a_{1}z+\cdots +a_n{z}^n\ne 0}$

for each complex number z. Let us define

${\displaystyle p^{\ast }(z)=z^{n}p\left(\frac{1}{z}\right)=a_0{z}^n+a_1{z}^{n-1}+\cdots +a_n.}$

Obviously, p*(z) ≠ 0 for all z in C. Consider the polynomial f(z) = p(z)p*(z). Then f(z) ≠ 0 for each z in C. Furthermore,

${\displaystyle f(\frac{1}{w})=p\left(\frac{1}{w}\right)p^{\ast }\left(\frac{1}{w}\right)=w^{-2n}{p}^{\ast }(w)p(w)=w^{-2n}f(w).}$

We can use this functional equation to prove that g, given by

${\displaystyle g=\frac{1}{|f(w){|}^{\frac{2}{n}}}|dw{|}^2}$

for w in C, and

${\displaystyle g=\frac{1}{{\left|f\left(\frac{1}{w}\right)\right|}^{\frac{2}{n}}}{\left|d\left(\frac{1}{w}\right)\right|}^2}$

for w ∈ S²\{0}, is a well defined Riemannian metric over the sphere S² (which we identify with the extended complex plane C ∪ {∞}).

Now, a simple computation shows that

${\displaystyle \mathrm{\forall }w\in \mathbf{C}:\frac{1}{|f(w){|}^{\frac{1}{n}}}{K}_g=\frac{1}{n}\mathrm{\Delta }\log |f(w)|=\frac{1}{n}\mathrm{\Delta }\text{Re}(\log f(w))=0,}$

since the real part of an analytic function is harmonic. This proves that K_g = 0.

Corollaries

Since the fundamental theorem of algebra can be seen as the statement that the field of complex numbers is algebraically closed, it follows that any theorem concerning algebraically closed fields applies to the field of complex numbers. Here are a few more consequences of the theorem, which are either about the field of real numbers or the relationship between the field of real numbers and the field of complex numbers:

• The field of complex numbers is the algebraic closure of the field of real numbers.
• Every polynomial in one variable z with complex coefficients is the product of a complex constant and polynomials of the form z + a with a complex.
• Every polynomial in one variable x with real coefficients can be uniquely written as the product of a constant, polynomials of the form x + a with a real, and polynomials of the form x² + ax + b with a and b real and a² − 4b < 0 (which is the same thing as saying that the polynomial x² + ax + b has no real roots). (By the Abel–Ruffini theorem, the real numbers a and b are not necessarily expressible in terms of the coefficients of the polynomial, the basic arithmetic operations and the extraction of n-th roots.) This implies that the number of non-real complex roots is always even and remains even when counted with their multiplicity.
• Every rational function in one variable x, with real coefficients, can be written as the sum of a polynomial function with rational functions of the form a/(x − b)ⁿ (where n is a natural number, and a and b are real numbers), and rational functions of the form (ax + b)/(x² + cx + d)ⁿ (where n is a natural number, and a, b, c, and d are real numbers such that c² − 4d < 0). A corollary of this is that every rational function in one variable and real coefficients has an elementary primitive.
• Every algebraic extension of the real field is isomorphic either to the real field or to the complex field.

Bounds on the zeros of a polynomial

Main article: Properties of polynomial roots

While the fundamental theorem of algebra states a general existence result, it is of some interest, both from the theoretical and from the practical point of view, to have information on the location of the zeros of a given polynomial. The simpler result in this direction is a bound on the modulus: all zeros ζ of a monic polynomial ${\displaystyle z^n+a_{n-1}{z}^{n-1}+\cdots +a_{1}z+a_0}$ satisfy an inequality |ζ| ≤ R_∞, where

${\displaystyle R_{\mathrm{\infty }}:=1+max\{|a_0|,\dots ,|a_{n-1}|\}.}$

As stated, this is not yet an existence result but rather an example of what is called an a priori bound: it says that if there are solutions then they lie inside the closed disk of center the origin and radius R_∞. However, once coupled with the fundamental theorem of algebra it says that the disk contains in fact at least one solution. More generally, a bound can be given directly in terms of any p-norm of the n-vector of coefficients ${\displaystyle a:=(a_0,a_1,\dots ,a_{n-1}),}$ that is |ζ| ≤ R_p, where R_p is precisely the q-norm of the 2-vector ${\displaystyle (1,\Vert a{\Vert }_p),}$ q being the conjugate exponent of p, ${\displaystyle \frac{1}{p}+\frac{1}{q}=1,}$ for any 1 ≤ p ≤ ∞. Thus, the modulus of any solution is also bounded by

${\displaystyle R_1:=max\left\{1,\sum _{0\le k<n}|a_k|\right\},}$

${\displaystyle R_p:={\left[1+{\left(\sum _{0\le k<n}|a_k{|}^p\right)}^{\frac{q}{p}}\right]}^{\frac{1}{q}},}$

for 1 < p < ∞, and in particular

${\displaystyle R_2:=\sqrt{\sum _{0\le k\le n}|a_k{|}^2}}$

(where we define a_n to mean 1, which is reasonable since 1 is indeed the n-th coefficient of our polynomial). The case of a generic polynomial of degree n,

${\displaystyle P(z):=a_n{z}^n+a_{n-1}{z}^{n-1}+\cdots +a_{1}z+a_0,}$

is of course reduced to the case of a monic, dividing all coefficients by a_n ≠ 0. Also, in case that 0 is not a root, i.e. a₀ ≠ 0, bounds from below on the roots ζ follow immediately as bounds from above on ${\displaystyle \frac{1}{\zeta }}$, that is, the roots of

${\displaystyle a_0{z}^n+a_1{z}^{n-1}+\cdots +a_{n-1}z+a_n.}$

Finally, the distance ${\displaystyle |\zeta -{\zeta }_0|}$ from the roots ζ to any point ${\displaystyle {\zeta }_0}$ can be estimated from below and above, seeing ${\displaystyle \zeta -{\zeta }_0}$ as zeros of the polynomial ${\displaystyle P(z+{\zeta }_0)}$, whose coefficients are the Taylor expansion of P(z) at ${\displaystyle z={\zeta }_0.}$

Let ζ be a root of the polynomial

${\displaystyle z^n+a_{n-1}{z}^{n-1}+\cdots +a_{1}z+a_0;}$

in order to prove the inequality |ζ| ≤ R_p we can assume, of course, |ζ| > 1. Writing the equation as

${\displaystyle -{\zeta }^n=a_{n-1}{\zeta }^{n-1}+\cdots +a_1\zeta +a_0,}$

and using the Hölder's inequality we find

${\displaystyle |\zeta {|}^n\le \Vert a{\Vert }_p{\Vert \left({\zeta }^{n-1},\dots ,\zeta ,1\right)\Vert }_q.}$

Now, if p = 1, this is

${\displaystyle |\zeta {|}^n\le \Vert a{\Vert }_{1}max\left\{|\zeta {|}^{n-1},\dots ,|\zeta |,1\right\}=\Vert a{\Vert }_1|\zeta {|}^{n-1},}$

thus

${\displaystyle |\zeta |\le max\{1,\Vert a{\Vert }_1\}.}$

In the case 1 < p ≤ ∞, taking into account the summation formula for a geometric progression, we have

${\displaystyle |\zeta {|}^n\le \Vert a{\Vert }_p{\left(|\zeta {|}^{q(n-1)}+\cdots +|\zeta {|}^q+1\right)}^{\frac{1}{q}}=\Vert a{\Vert }_p{\left(\frac{|\zeta {|}^{qn}-1}{|\zeta {|}^q-1}\right)}^{\frac{1}{q}}\le \Vert a{\Vert }_p{\left(\frac{|\zeta {|}^{qn}}{|\zeta {|}^q-1}\right)}^{\frac{1}{q}},}$

thus

${\displaystyle |\zeta {|}^{nq}\le \Vert a{\Vert }_p^q\frac{|\zeta {|}^{qn}}{|\zeta {|}^q-1}}$

and simplifying,

${\displaystyle |\zeta {|}^q\le 1+\Vert a{\Vert }_p^q.}$

Therefore

${\displaystyle |\zeta |\le {\Vert \left(1,\Vert a{\Vert }_p\right)\Vert }_q=R_p}$

holds, for all 1 ≤ p ≤ ∞.

Swamp

From Wikipedia, the free encyclopedia
https://en.wikipedia.org/wiki/Swamp

A freshwater swamp in Florida, United States

A swamp is a forested wetland. Swamps are considered to be transition zones because both land and water play a role in creating this environment. Swamps vary in size and are located all around the world. The water of a swamp may be fresh water, brackish water, or seawater. Freshwater swamps form along large rivers or lakes where they are critically dependent upon rainwater and seasonal flooding to maintain natural water level fluctuations. Saltwater swamps are found along tropical and subtropical coastlines. Some swamps have hammocks, or dry-land protrusions, covered by aquatic vegetation, or vegetation that tolerates periodic inundation or soil saturation. The two main types of swamp are "true" or swamp forests and "transitional" or shrub swamps. In the boreal regions of Canada, the word swamp is colloquially used for what is more formally termed a bog, fen, or muskeg. Some of the world's largest swamps are found along major rivers such as the Amazon, the Mississippi, and the Congo.

Differences between marshes and swamps

Difference between swamp and marsh

Swamps and marshes are specific types of wetlands that form along waterbodies containing rich, hydric soils. Marshes are wetlands, continually or frequently flooded by nearby running bodies of water, that are dominated by emergent soft-stem vegetation and herbaceous plants. Swamps are wetlands consisting of saturated soils or standing water and are dominated by water-tolerant woody vegetation such as shrubs, bushes, and trees.

Hydrology

Swamps are characterized by their saturated soils and slow-moving waters. The water that accumulates in swamps comes from a variety of sources including precipitation, groundwater, tides and/or freshwater flooding. These hydrologic pathways all contribute to how energy and nutrients flow in and out of the ecosystem. As water flows through the swamp, nutrients, sediment and pollutants are naturally filtered out. Chemicals like phosphorus and nitrogen that end up in waterways get absorbed and used by the aquatic plants within the swamp, purifying the water. Any remaining or excess chemicals present will accumulate at the bottom of the swamp, being removed from the water and buried within the sediment. The biogeochemical environment of a swamp is dependent on its hydrology, affecting the levels and availability of resources like oxygen, nutrients, water pH and toxicity, which will influence the whole ecosystem.

Values and ecosystem services

The Linnaistensuo Mire, a nature reserve swamp in Lahti, Finland.

Swamps and other wetlands have traditionally held a very low property value compared to fields, prairies, or woodlands. They have a reputation for being unproductive land that cannot easily be utilized for human activities, other than hunting, trapping, or fishing. Farmers, for example, typically drained swamps next to their fields so as to gain more land usable for planting crops, both historically, and to a lesser extent, presently. On the other hand, swamps can (and do) play a beneficial ecological role in the overall functions of the natural environment and provide a variety of resources that many species depend on. Swamps and other wetlands have shown to be a natural form of flood management and defense against flooding. In such circumstances where flooding does occur, swamps absorb and use the excess water within the wetland, preventing it from traveling and flooding surrounding areas. Dense vegetation within the swamp also provides soil stability to the land, holding soils and sediment in place whilst preventing erosion and land loss. Swamps are an abundant and valuable source of fresh water and oxygen for all life, and they are often breeding grounds for a wide variety of species. Floodplain swamps are an important resource in the production and distribution of fish. Two thirds of global fish and shellfish are commercially harvested and dependent on wetlands.

Impacts and conservation

Historically, humans have been known to drain and/or fill swamps and other wetlands in order to create more space for human development and to reduce the threat of diseases borne by swamp insects. Wetlands are removed and replaced with land that is then used for things like agriculture, real estate, and recreational uses. Many swamps have also undergone intensive logging and farming, requiring the construction of drainage ditches and canals. These ditches and canals contributed to drainage and, along the coast, allowed salt water to intrude, converting swamps to marsh or even to open water. Large areas of swamp were therefore lost or degraded. Louisiana provides a classic example of wetland loss from these combined factors. Europe has probably lost nearly half its wetlands. New Zealand lost 90 percent of its wetlands over a period of 150 years. Ecologists recognize that swamps provide ecological services including flood control, fish production, water purification, carbon storage, and wildlife habitats. In many parts of the world authorities protect swamps. In parts of Europe and North America, swamp restoration projects are becoming widespread. The United States government began enforcing stricter laws and management programs in the 1970s in efforts to protect and restore these ecosystems. Often the simplest steps to restoring swamps involve plugging drainage ditches and removing levees.

Conservationists work to preserve swamps such as those in northwest Indiana in the United States Midwest that were preserved as part of the Indiana Dunes.

Notable examples

Swamps can be found on all continents except Antarctica.

The largest swamp in the world is the Amazon River floodplain, which is particularly significant for its large number of fish and tree species.

Africa

The Sudd and the Okavango Delta are Africa's best known marshland areas. The Bangweulu Floodplains make up Africa's largest swamp.

Asia

Marsh Arabs poling a mashoof

The Mesopotamian Marshes is a large swamp and river system in southern Iraq, traditionally inhabited in part by the Marsh Arabs.

In Asia, tropical peat swamps are located in mainland East Asia and Southeast Asia. In Southeast Asia, peatlands are mainly found in low altitude coastal and sub-coastal areas and extend inland for distance more than 100 km (62 mi) along river valleys and across watersheds. They are mostly to be found on the coasts of East Sumatra, Kalimantan (Central, East, South and West Kalimantan provinces), West Papua, Papua New Guinea, Brunei, Peninsular Malaya, Sabah, Sarawak, Southeast Thailand, and the Philippines (Riley et al.,1996). Indonesia has the largest area of tropical peatland. Of the total 440,000 km² (170,000 sq mi) tropical peat swamp, about 210,000 km² (81,000 sq mi) are located in Indonesia (Page, 2001; Wahyunto, 2006).

The Vasyugan Swamp is a large swamp in the western Siberia area of the Russian Federation. This is one of the largest swamps in the world, covering an area larger than Switzerland.

North America

Swamp in southern Louisiana

The Atchafalaya Swamp at the lower end of the Mississippi River is the largest swamp in the United States. It is an important example of southern cypress swamp but it has been greatly altered by logging, drainage and levee construction. Other famous swamps in the United States are the forested portions of the Everglades, Okefenokee Swamp, Barley Barber Swamp, Great Cypress Swamp and the Great Dismal Swamp. The Okefenokee is located in extreme southeastern Georgia and extends slightly into northeastern Florida. The Great Cypress Swamp is mostly in Delaware but extends into Maryland on the Delmarva Peninsula. Point Lookout State Park on the southern tip of Maryland contains a large amount of swamps and marshes. The Great Dismal Swamp lies in extreme southeastern Virginia and extreme northeastern North Carolina. Both are National Wildlife Refuges. Another swamp area, Reelfoot Lake of extreme western Tennessee and Kentucky, was created by the 1811–12 New Madrid earthquakes. Caddo Lake, the Great Dismal and Reelfoot are swamps that are centered at large lakes. Swamps are often associated with bayous in the southeastern United States, especially in the Gulf Coast region. A baygall is a type of swamp found in the forest of the Gulf Coast states in the USA.

List of major swamps

A small swamp in Padstow, New South Wales, Australia

Inside a mangrove canopy, Salt Pan Creek, New South Wales

The world's largest wetlands include significant areas of swamp, such as in the Amazon and Congo River basins. Further north, however, the largest wetlands are bogs.

Africa

• Bangweulu Swamps, Zambia
• Mare aux Songes, Mauritius*
• Niger Delta, Nigeria
• Okavango Swamp, Botswana
• Sudd, South Sudan

Asia

• Asmat Swamp, Indonesia
• Candaba Swamp in Apalit and Candaba, Pampanga and Pulilan, Bulacan, Philippines
• Mangrove Swamp in Karachi, Pakistan
• Myristica Swamp in Western Ghats, India
• Ratargul Swamp Forest in Sylhet, Bangladesh
• Sundarbans in India and Bangladesh
• Vasyugan Swamp, Russia
• Negombo Swamp, Sri Lanka

Australia

• Banksia Swamp, Victoria, Australia
• Becher Point Wetlands, Western Australia
• Burraga Swamp, New South Wales, Australia
• Coomonderry Swamp
• Coastal Swamp Oak Forest, Queensland/New South Wales, Australia
• Coastal Upland Swamps, New South Wales, Australia
• Cumbung Swamp, New South Wales, Australia
• Fivebough and Tuckerbil Swamps, New South Wales, Australia
• Koo-Wee-Rup Swamp, Victoria, Australia
• Noosa Everglades, Queensland, Australia
• Toolibin Lake, Western Australia
• West Melbourne Swamp, Victoria, Australia

Europe

A black alder swamp in Germany

• Pripyat Marshes, Belarus
• Šúr, Slovakia

North America

• Atchafalaya National Wildlife Refuge, Louisiana, United States
• Big Cypress National Preserve, Florida, United States
• Barley Barber Swamp, Florida, United States
• Cache River, Illinois, United States
• Caddo Lake, Texas/Louisiana, United States
• Cibuco Swamp, Puerto Rico
• Congaree Swamp, South Carolina, United States
• Everglades, Florida, United States
• First Landing State Park, Virginia, United States
• Grand Kankakee Marsh, Indiana, United States
• Great Black Swamp, Indiana/Ohio, United States
• Great Cypress Swamp, Delaware and Maryland, United States, also known as Great Pocomoke Swamp
• Great Dismal Swamp, North Carolina/Virginia, United States
• Great Swamp National Wildlife Refuge, New Jersey, United States
• Green Swamp, Florida, United States
• Green Swamp, North Carolina, United States
• Honey Island Swamp, Louisiana, United States
• Hudson Bay Lowlands, Ontario, Canada
• Limberlost, Indiana, United States
• Louisiana swamplands, Louisiana, United States
• Mingo National Wildlife Refuge, Puxico, Missouri, United States
• Okefenokee Swamp, Georgia/Florida, United States
• Pantanos de Centla, Tabasco/Campeche, Mexico
• Reelfoot Lake, Tennessee/Kentucky, United States
• Texas Swamplands, Texas, United States
• Tortuguero National Park, Limón, Costa Rica

South America

Pantanal in Brazil

• Caribbean Lowlands, Colombia
• Esteros del Iberá, Argentina
• Lahuen Ñadi, Chile
• Pantanal, Brazil, Bolivia and Paraguay
• Paraná Delta, Argentina

Sex-determination system

From Wikipedia, the free encyclopedia
https://en.wikipedia.org/wiki/Sex-determination_system

Some chromosomal sex determination systems in animals

A sex-determination system is a biological system that determines the development of sexual characteristics in an organism. Most organisms that create their offspring using sexual reproduction have two common sexes and a few less common intersex variations.

In some species there are hermaphrodites. There are also some species that are only one sex due to parthenogenesis, the act of a female reproducing without fertilization.

In some species, sex determination is genetic: males and females have different alleles or even different genes that specify their sexual morphology. In animals this is often accompanied by chromosomal differences, generally through combinations of XY, ZW, XO, ZO chromosomes, or haplodiploidy. The sexual differentiation is generally triggered by a main gene (a "sex locus"), with a multitude of other genes following in a domino effect.

In other cases, sex of a fetus is determined by environmental variables (such as temperature). The details of some sex-determination systems are not yet fully understood. Hopes for future fetal biological system analysis include complete-reproduction-system initialized signals that can be measured during pregnancies to more accurately determine whether a determined sex of a fetus is male, or female. Such analysis of biological systems could also signal whether the fetus is hermaphrodite, which includes total or partial of both male and female reproduction organs.

Some species such as various plants and fish do not have a fixed sex, and instead go through life cycles and change sex based on genetic cues during corresponding life stages of their type. This could be due to environmental factors such as seasons and temperature. In some gonochoric species, a few individuals may have sex characteristics of both sexes, a condition called intersex.

While diversity in sex determination systems is common throughout different biological systems, the systems beyond XY/XX/XO in mammals are often left to more advanced courses for those whose studies specialize in genetics of other organisms.

Discovery

Sex determination was discovered in the mealworm by the American geneticist Nettie Stevens in 1903.

Chromosomal systems

XX/XY sex chromosomes

Drosophila sex-chromosomes

Human male XY chromosomes after G-banding

Main article: XY sex-determination system

The XX/XY sex-determination system is the most familiar, as it is found in humans. The XX/XY system is found in most other mammals, as well as some insects. In this system, most females have two of the same kind of sex chromosome (XX), while most males have two distinct sex chromosomes (XY). The X and Y sex chromosomes are different in shape and size from each other, unlike the rest of the chromosomes (autosomes), and are sometimes called allosomes. In some species, such as humans, organisms remain sex indifferent for a time after they're created; in others, however, such as fruit flies, sexual differentiation occurs as soon as the egg is fertilized.

Y-centered sex determination

Some species (including humans) have a gene SRY on the Y chromosome that determines maleness. Members of SRY-reliant species can have uncommon XY chromosomal combinations such as XXY and still live. Human sex is determined by the presence or absence of a Y chromosome with a functional SRY gene. Once the SRY gene is activated, cells create testosterone and anti-müllerian hormone which typically ensures the development of a single, male reproductive system. In typical XX embryos, cells secrete estrogen, which drives the body toward the female pathway.

In Y-centered sex determination, the SRY gene is the main gene in determining male characteristics, but multiple genes are required to develop testes. In XY mice, lack of the gene DAX1 on the X chromosome results in sterility, but in humans it causes adrenal hypoplasia congenita. However, when an extra DAX1 gene is placed on the X chromosome, the result is a female, despite the existence of SRY. Even when there are normal sex chromosomes in XX females, duplication or expression of SOX9 causes testes to develop. Gradual sex reversal in developed mice can also occur when the gene FOXL2 is removed from females. Even though the gene DMRT1 is used by birds as their sex locus, species who have XY chromosomes also rely upon DMRT1, contained on chromosome 9, for sexual differentiation at some point in their formation.

X-centered sex determination

Some species, such as fruit flies, use the presence of two X chromosomes to determine femaleness. Species that use the number of Xs to determine sex are nonviable with an extra X chromosome.

Other variants of XX/XY sex determination

Some fish have variants of the XY sex-determination system, as well as the regular system. For example, while having an XY format, Xiphophorus nezahualcoyotl and X. milleri also have a second Y chromosome, known as Y', that creates XY' females and YY' males.

At least one monotreme, the platypus, presents a particular sex determination scheme that in some ways resembles that of the ZW sex chromosomes of birds and lacks the SRY gene. The platypus has ten sex chromosomes; males have an XYXYXYXYXY pattern while females have ten X chromosomes. Although it is an XY system, the platypus' sex chromosomes share no homologues with eutherian sex chromosomes. Instead, homologues with eutherian sex chromosomes lie on the platypus chromosome 6, which means that the eutherian sex chromosomes were autosomes at the time that the monotremes diverged from the therian mammals (marsupials and eutherian mammals). However, homologues to the avian DMRT1 gene on platypus sex chromosomes X3 and X5 suggest that it is possible the sex-determining gene for the platypus is the same one that is involved in bird sex-determination. More research must be conducted in order to determine the exact sex determining gene of the platypus.

Heredity of sex chromosomes in XO sex determination

XX/X0 sex chromosomes

Main article: X0 sex-determination system

In this variant of the XY system, females have two copies of the sex chromosome (XX) but males have only one (X0). The 0 denotes the absence of a second sex chromosome. Generally in this method, the sex is determined by amount of genes expressed across the two chromosomes. This system is observed in a number of insects, including the grasshoppers and crickets of order Orthoptera and in cockroaches (order Blattodea). A small number of mammals also lack a Y chromosome. These include the Amami spiny rat (Tokudaia osimensis) and the Tokunoshima spiny rat (Tokudaia tokunoshimensis) and Sorex araneus, a shrew species. Transcaucasian mole voles (Ellobius lutescens) also have a form of XO determination, in which both sexes lack a second sex chromosome. The mechanism of sex determination is not yet understood.

The nematode C. elegans is male with one sex chromosome (X0); with a pair of chromosomes (XX) it is a hermaphrodite. Its main sex gene is XOL, which encodes XOL-1 and also controls the expression of the genes TRA-2 and HER-1. These genes reduce male gene activation and increase it, respectively.

ZW/ZZ sex chromosomes

Main article: ZW sex-determination system

The ZW sex-determination system is found in birds, some reptiles, and some insects and other organisms. The ZW sex-determination system is reversed compared to the XY system: females have two different kinds of chromosomes (ZW), and males have two of the same kind of chromosomes (ZZ). In the chicken, this was found to be dependent on the expression of DMRT1. In birds, the genes FET1 and ASW are found on the W chromosome for females, similar to how the Y chromosome contains SRY. However, not all species depend upon the W for their sex. For example, there are moths and butterflies that are ZW, but some have been found female with ZO, as well as female with ZZW. Also, while mammals deactivate one of their extra X chromosomes when female, it appears that in the case of Lepidoptera, the males produce double the normal amount of enzymes, due to having two Z's. Because the use of ZW sex determination is varied, it is still unknown how exactly most species determine their sex. However, reportedly, the silkworm Bombyx mori uses a single female-specific piRNA as the primary determiner of sex. Despite the similarities between the ZW and XY systems, these sex chromosomes evolved separately. In the case of the chicken, their Z chromosome is more similar to humans' autosome 9. The chicken's Z chromosome also seems to be related to the X chromosome of the platypus. When a ZW species, such as the Komodo dragon, reproduces parthenogenetically, usually only males are produced. This is due to the fact that the haploid eggs double their chromosomes, resulting in ZZ or WW. The ZZ become males, but the WW are not viable and are not brought to term.

In both XY and ZW sex determination systems, the sex chromosome carrying the critical factors is often significantly smaller, carrying little more than the genes necessary for triggering the development of a given sex.

ZZ/Z0 sex chromosomes

Main article: Z0 sex-determination system

The ZZ/Z0 sex-determination system is found in some moths. In these insects there is one sex chromosome, Z. Males have two Z chromosomes, whereas females have one Z. Males are ZZ, while females are Z0.

UV sex chromosomes

In some Bryophyte and some algae species, the gametophyte stage of the life cycle, rather than being hermaphrodite, occurs as separate male or female individuals that produce male and female gametes respectively. When meiosis occurs in the sporophyte generation of the life cycle, the sex chromosomes known as U and V assort in spores that carry either the U chromosome and give rise to female gametophytes, or the V chromosome and give rise to male gametophytes.

Haplodiploid sex chromosomes

Haplodiploidy

Main article: Haplodiploidy

Haplodiploidy is found in insects belonging to Hymenoptera, such as ants and bees. Sex determination is controlled by the zygosity of a complementary sex determiner (csd) locus. Unfertilized eggs develop into haploid individuals which have a single, hemizygous copy of the csd locus and are therefore males. Fertilized eggs develop into diploid individuals which, due to high variability in the csd locus, are generally heterozygous females. In rare instances diploid individuals may be homozygous, these develop into sterile males. The gene acting as a csd locus has been identified in the honeybee and several candidate genes have been proposed as a csd locus for other Hymenopterans. Most females in the Hymenoptera order can decide the sex of their offspring by holding received sperm in their spermatheca and either releasing it into their oviduct or not. This allows them to create more workers, depending on the status of the colony.

Other chromosomal systems

Other uncommon systems include those of the green swordtail (a polyfactorial system with the sex-determining genes on several chromosomes) the Chironomus midges; the juvenile hermaphroditism of zebrafish, with an unknown trigger; and the platyfish, which has W, X, and Y chromosomes. This allows WY, WX, or XX females and YY or XY males.

Mating type in microorganisms is analogous to sex in multi-cellular organisms, and is sometimes described using those terms, though they are not necessarily correlated with physical body structures. Some species have more than two mating types. Tetrahymena, a type of ciliate, has seven mating types. Schizophyllum commune, a type of fungus, has 23,328.

Environmental systems

Main article: Environmental sex determination

All alligators determine the sex of their offspring by the temperature of the nest.

Temperature-dependent

Main article: Temperature-dependent sex determination

Many other sex-determination systems exist. In some species of reptiles, including alligators, some turtles, and the tuatara, sex is determined by the temperature at which the egg is incubated during a temperature-sensitive period. There are no examples of temperature-dependent sex determination (TSD) in birds. Megapodes had formerly been thought to exhibit this phenomenon, but were found to actually have different temperature-dependent embryo mortality rates for each sex. For some species with TSD, sex determination is achieved by exposure to hotter temperatures resulting in the offspring being one sex and cooler temperatures resulting in the other. This type of TSD is called Pattern I. For others species using TSD, it is exposure to temperatures on both extremes that results in offspring of one sex, and exposure to moderate temperatures that results in offspring of the opposite sex, called Pattern II TSD. The specific temperatures required to produce each sex are known as the female-promoting temperature and the male-promoting temperature. When the temperature stays near the threshold during the temperature sensitive period, the sex ratio is varied between the two sexes. Some species' temperature standards are based on when a particular enzyme is created. These species that rely upon temperature for their sex determination do not have the SRY gene, but have other genes such as DAX1, DMRT1, and SOX9 that are expressed or not expressed depending on the temperature. The sex of some species, such as the Nile tilapia, Australian skink lizard, and Australian dragon lizard, has an initial bias, set by chromosomes, but can later be changed by the temperature of incubation.

It is unknown how exactly temperature-dependent sex determination evolved. It could have evolved through certain sexes being more suited to certain areas that fit the temperature requirements. For example, a warmer area could be more suitable for nesting, so more females are produced to increase the amount that nest next season. In amniotes, environmental sex determination preceded the genetically determined systems of birds and mammals; it is thought that a temperature-dependent amniote was the common ancestor of amniotes with sex chromosomes.

Other environmental systems

There are other environmental sex determination systems including location-dependent determination systems as seen in the marine worm Bonellia viridis – larvae become males if they make physical contact with a female, and females if they end up on the bare sea floor. This is triggered by the presence of a chemical produced by the females, bonellin. Some species, such as some snails, practice sex change: adults start out male, then become female. In tropical clown fish, the dominant individual in a group becomes female while the other ones are male, and bluehead wrasses (Thalassoma bifasciatum) are the reverse. Some species, however, have no sex-determination system. Hermaphrodite species include the common earthworm and certain species of snails. A few species of fish, reptiles, and insects reproduce by parthenogenesis and are female altogether. There are some reptiles, such as the boa constrictor and Komodo dragon that can reproduce both sexually and asexually, depending on whether a mate is available.

Evolution

The ends of the XY chromosomes, highlighted here in green, are all that is left of the original autosomes that can still cross over with each other.

Sex determination systems may have evolved from mating type, which is a feature of microorganisms.

Chromosomal sex determination may have evolved early in the history of eukaryotes. But in plants it has been suggested to have evolved recently.

The accepted hypothesis of XY and ZW sex chromosome evolution in amniotes is that they evolved at the same time, in two different branches.

No genes are shared between the avian ZW and mammal XY chromosomes and the chicken Z chromosome is similar to the human autosomal chromosome 9, rather than X or Y. This suggests not that the ZW and XY sex-determination systems share an origin but that the sex chromosomes are derived from autosomal chromosomes of the common ancestor of birds and mammals. In the platypus, a monotreme, the X₁ chromosome shares homology with therian mammals, while the X₅ chromosome contains an avian sex-determination gene, further suggesting an evolutionary link.

However, there is some evidence to suggest that there could have been transitions between ZW and XY, such as in Xiphophorus maculatus, which have both ZW and XY systems in the same population, despite the fact that ZW and XY have different gene locations. A recent theoretical model raises the possibility of both transitions between the XY/XX and ZZ/ZW system and environmental sex determination. The platypus' genes also back up the possible evolutionary link between XY and ZW, because they have the DMRT1 gene possessed by birds on their X chromosomes. Regardless, XY and ZW follow a similar route. All sex chromosomes started out as an original autosome of an original amniote that relied upon temperature to determine the sex of offspring. After the mammals separated, the reptile branch further split into Lepidosauria and Archosauromorpha. These two groups both evolved the ZW system separately, as evidenced by the existence of different sex chromosomal locations. In mammals, one of the autosome pair, now Y, mutated its SOX3 gene into the SRY gene, causing that chromosome to designate sex. After this mutation, the SRY-containing chromosome inverted and was no longer completely homologous with its partner. The regions of the X and Y chromosomes that are still homologous to one another are known as the pseudoautosomal region. Once it inverted, the Y chromosome became unable to remedy deleterious mutations, and thus degenerated. There is some concern that the Y chromosome will shrink further and stop functioning in ten million years: but the Y chromosome has been strictly conserved after its initial rapid gene loss.

There are some vertebrate species, such as the medaka fish, that evolved sex chromosomes separately; their Y chromosome never inverted and can still swap genes with the X. These species' sex chromosomes are relatively primitive and unspecialized. Because the Y does not have male-specific genes and can interact with the X, XY and YY females can be formed as well as XX males. Non-inverted Y chromosomes with long histories are found in pythons and emus, each system being more than 120 million years old, suggesting that inversions are not necessarily an eventuality. XO sex determination can evolve from XY sex determination with about 2 million years.