[Paper Review] A Wronskian Approach to the real τ-conjecture

Pascal Koiran, Natacha Portier+1 more|HAL (Le Centre pour la Communication Scientifique Directe)|May 4, 2012

Complexity and Algorithms in Graphs3 citations

TL;DR

This paper presents a Wronskian-based approach to bound the number of real roots of sums of products of sparse univariate polynomials, achieving a polynomial-in-t, quasi-polynomial-in-k and exponential-in-m upper bound. It resolves a key case of the real τ-conjecture by removing doubly exponential dependence on k, and provides a deterministic polynomial identity testing algorithm for the same class of polynomials with improved complexity over prior work.

ABSTRACT

According to the real τ-conjecture, the number of real roots of a sum of products of sparse polynomials should be polynomially bounded in the size of such an expression. It is known that this conjecture implies a superpolynomial lower bound on the arithmetic circuit complexity of the permanent. In this paper, we use the Wronksian determinant to give an upper bound on the number of real roots of sums of products of sparse polynomials. The proof technique is quite versatile; it can in particular be applied to some sparse geometric problems that do not originate from arithmetic circuit complexity. The paper should therefore be of interest to researchers from these two communities (complexity theory and sparse polynomial systems).

Motivation & Objective

• To address the real τ-conjecture by bounding the number of real roots of sums of products of sparse univariate polynomials.
• To improve upon prior exponential and doubly exponential bounds in k and m by achieving a t^{O(k²m)} upper bound on real roots.
• To develop a deterministic polynomial identity testing (PIT) algorithm for the same class of polynomials with reduced dependence on k and m.
• To extend the applicability of Wronskian and Descartes' rule techniques beyond arithmetic circuit complexity to sparse geometric problems.
• To establish the optimality of the root bound via a construction showing tightness up to constants.

Proposed method

• Utilizes the Wronskian determinant of functions f^{α_i + k} to analyze linear dependence and root structure of sums of powers of sparse polynomials.
• Applies a weak form of Descartes' rule of signs to bound positive real roots by t−1 for t-sparse polynomials with real exponents.
• Employs Rolle’s theorem and intermediate value theorem to construct examples with many roots, demonstrating tightness of bounds.
• Introduces a key lemma linking the vanishing of Wronskians to the zeros of f f′, enabling control over the number of critical points.
• Uses composition of functions g = h ∘ f with h having many roots to construct polynomials with many real roots.
• Develops a deterministic PIT algorithm by combining Wronskian analysis with root counting and coefficient evaluation, achieving exponential dependence only in k and m.

Experimental results

Research questions

• RQ1 Can the number of real roots of sums of products of sparse polynomials be bounded polynomially in t and quasi-polynomially in k, with only exponential dependence on m?
• RQ2 Does the Wronskian determinant provide a sufficient tool to control the number of real roots in sums of powers of sparse polynomials?
• RQ3 Can the bound on real roots be improved from doubly exponential in k to singly exponential, as required for implications to the permanent's circuit complexity?
• RQ4 Is there a deterministic polynomial identity testing algorithm for this class of polynomials with complexity polynomial in t and exponential in k and m?
• RQ5 Can the Wronskian-based approach be generalized to sparse geometric problems beyond arithmetic circuit complexity?

Key findings

• The number of real roots of a sum of the form ∑_{i=1}^k ∏_{j=1}^m f_j^{α_{i,j}} is bounded by t^{O(k²m)}, removing the doubly exponential dependence on k present in prior work.
• The bound is polynomial in t, exponential in m, and quasi-polynomial in k, representing a significant improvement over previous results.
• A deterministic polynomial identity testing algorithm is developed with running time polynomial in t and the bit size of coefficients and exponents, and exponential in k and m.
• The paper proves that the Wronskian-based root bound is optimal up to constants, via a construction showing that Z(g) ≥ (|ϒ| + 1)(k − 1) + Z(f) for some g.
• The method extends to non-integer exponents and provides a polynomial upper bound for the wider class of functions ∑_{i=1}^k x^{α_i}(ax + b)^{β_i} with real α_i, β_i.
• The approach gives a new proof of the weak Descartes' rule of signs and generalizes results from Avendaño on sparse curve-line intersections.