In mathematics, Cohn's theorem states that a ''n''th-degree self-inversive

polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An ex ...

p(z)

has as many roots in the open unit disk

D =\

as the

reciprocal polynomial In algebra, given a polynomial :p(x) = a_0 + a_1x + a_2x^2 + \cdots + a_nx^n, with coefficients from an arbitrary field, its reciprocal polynomial or reflected polynomial,* denoted by or , is the polynomial :p^*(x) = a_n + a_x + \cdots + a_0x^n ...

of its

derivative In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...

. Cohn's theorem is useful for studying the distribution of the roots of self-inversive and self-reciprocal polynomials in the

complex plane In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by th ...

. An ''n''th-degree polynomial, :

p(z) = p_0 + p_1 z + \cdots + p_n z^n

is called self-inversive if there exists a ''fixed''

complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...

(

\omega

) of modulus 1 so that, :

p(z) = \omega  p^*(z),\qquad \left(, \omega, =1\right),

where :

p^*(z)=z^n \bar\left(1 / \bar\right) =\bar_n + \bar_ z + \cdots + \bar_0 z^n

is the

associated with

p(z)

and the bar means

complex conjugation In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, (if a and b are real, then) the complex conjugate of a + bi is equal to a ...

. Self-inversive polynomials have many interesting properties. For instance, its roots are all

symmetric Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definit ...

with respect to the

unit circle In mathematics, a unit circle is a circle of unit radius—that is, a radius of 1. Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system in the Eucli ...

and a polynomial whose roots are all on the unit circle is necessarily self-inversive. The

coefficient In mathematics, a coefficient is a multiplicative factor in some term of a polynomial, a series, or an expression; it is usually a number, but may be any expression (including variables such as , and ). When the coefficients are themselves ...

s of self-inversive polynomials satisfy the relations. :

p_k = \omega \bar_, \qquad 0 \leqslant k \leqslant n.

In the case where

\omega = 1,

a ''self-inversive polynomial'' becomes a ''complex-reciprocal polynomial'' (also known as a ''self-conjugate polynomial''). If its coefficients are real then it becomes a ''real self-reciprocal polynomial''. The ''

formal derivative In mathematics, the formal derivative is an operation on elements of a polynomial ring or a ring of formal power series that mimics the form of the derivative from calculus. Though they appear similar, the algebraic advantage of a formal deriva ...

'' of

p(z)

is a (''n'' − 1)th-degree polynomial given by :

q(z) =p'(z) =  p_1 + 2p_2 z + \cdots + n p_n z^.

Therefore, Cohn's theorem states that both

p(z)

and the polynomial :

q^*(z) =z^\bar_\left(1 / \bar\right) = z^ \bar' \left(1 / \bar\right) = n \bar_n + (n-1)\bar_ z + \cdots + \bar_1 z^

have the same number of roots in

, z, <1.

References

{{reflist Theorems about polynomials