The B. and M. Shapiro conjecture in real algebraic geometry is true

Algebra, Analysis 1 Minute

You need to know: Set {\mathbb C} of complex numbers, polynomials in complex variable, root of a polynomial, derivative, notation P^{(j)}(z) for j-th derivative of polynomial P(z), matrix, determinant of a matrix, vector space, basis of a vector space.

Background: Let {\cal P}=\{P_1(z), P_2(z), \dots, P_k(z)\} be a finite set of polynomials in complex variable z with complex coefficients. The complex span S of {\cal P} is the set of polynomials P(z) of the form P(z)=\sum\limits_{i=1}^k \lambda_i P_i(z) for some \lambda_i \in {\mathbb C}, \, i=1,\dots,k. The Wronskian W(z) of {\cal P} is the determinant of the k \times k matrix with entries P^{(j-1)}_i(z), \, i=1,\dots,k, \, j=1,\dots,k.

The Theorem: On 14th December 2005, Evgeny Mukhin, Vitaly Tarasov, and Alexander Varchenko submitted to arxiv a paper in which they proved that if all roots of the Wronskian of a set {\cal P} of polynomials are real, then the complex span of {\cal P} has a basis consisting of polynomials with real coefficients.

Short context: The Theorem confirms the 1995 conjecture known as the B. and M. Shapiro conjecture in real algebraic geometry. Earlier, it was known only in the case k=2.

Links: Free arxiv version of the original paper is here, journal version is here. See also Section 9.10 of this book for an accessible description of the Theorem.

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Published by Bogdan Grechuk

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  1. Pingback: All critical points of a rational function f are real only if f is equivalent to real – Theorems of the 21st century

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