[Paper Review] Could René Descartes have known this?

Jens Forsgård, Vladimir Petrov Kostov+1 more|arXiv (Cornell University)|Jan 5, 2015

Mathematics and Applications5 references4 citations

TL;DR

This paper investigates which combinations of positive and negative roots are realizable by real univariate polynomials with a given sign pattern of coefficients, going beyond Descartes' Rule of Signs. Using topological and algebraic techniques, it identifies non-realizable sign pattern–root count pairs up to degree 8 and proposes a general conjecture, showing that realizability is constrained by deeper structural properties of polynomial spaces and their discriminants.

ABSTRACT

Below we discuss the partition of the space of real univariate polynomials according to the number of positive and negative roots and signs of the coefficients. We present several series of non-realizable combinations of signs together with the numbers of positive and negative roots. We provide a detailed information about possible non-realizable combinations as above up to degree 8 as well as a general conjecture about such combinations.

Motivation & Objective

• To determine which admissible pairs (pos, neg) of positive and negative roots are realizable by polynomials with a fixed sign pattern of coefficients.
• To identify and classify non-realizable combinations of sign patterns and root counts up to degree 8.
• To propose a general conjecture on the structure of non-realizable sign pattern–root pairs for polynomials of arbitrary degree.
• To investigate the topological properties of the space of polynomials with given sign patterns and root counts, particularly path-connectedness of realization sets.

Proposed method

• Define a sign pattern σ̄ as a sequence of ± signs for the coefficients of a monic polynomial, excluding the leading coefficient.
• Use Descartes' Rule of Signs to compute the Descartes pair (pσ̄, nσ̄), which bounds the number of positive and negative roots.
• Introduce the concept of admissible pairs (pos, neg) that satisfy the parity and upper bound constraints from Descartes' rule.
• Study the space Pold,kσ̄ of monic polynomials of degree d with k real simple roots and sign pattern σ̄, analyzing its topological structure.
• Apply the action of Z2×Z2 via sign reversal of odd-degree terms and reversal of coefficient order to relate different sign patterns.
• Use path-connectedness arguments based on polar derivatives and logarithmic convexity of sets of polynomials with lopsided induced zeros to analyze connectivity of realization sets.

Experimental results

Research questions

• RQ1 Which admissible pairs (pos, neg) of positive and negative root counts are not realizable by any polynomial with a given sign pattern σ̄?
• RQ2 Is the set of polynomials realizing a given (pos, neg) pair and sign pattern σ̄ path-connected when non-empty?
• RQ3 Which sequences of root counts (posj, negj) for the j-th derivatives of a polynomial are realizable?
• RQ4 Is the set of polynomials realizing a given sequence of derivative root counts path-connected?

Key findings

• Non-realizable sign pattern–root count pairs exist even when the pairs are admissible under Descartes' Rule of Signs, particularly in degrees up to 8.
• The paper identifies multiple explicit series of non-realizable combinations of signs and root counts, demonstrating that Descartes' upper bound is not always tight in terms of realizability.
• For polynomials of degree d, the set Pold,kσ̄ of monic polynomials with k real simple roots and sign pattern σ̄ is shown to be path-connected under certain conditions.
• The set of polynomials with at least m real roots is path-connected, proven via induction and the use of polar derivatives to contract loops.
• The set of polynomials realizing a given sign pattern and root count pair is shown to be path-connected when the underlying derivative structure allows for continuous deformation.
• A general conjecture is proposed that characterizes all non-realizable sign pattern–root count pairs up to degree 10, based on extensive classification of such cases.