[Paper Review] Vandermonde varieties and relations among Schur polynomials

Ralf Fröberg, Boris Shapiro|arXiv (Cornell University)|Feb 6, 2013

Advanced Combinatorial Mathematics10 references3 citations

TL;DR

This paper introduces Vandermonde varieties—algebraic structures tied to linear recurrence relations with prescribed zero sets—and establishes their free resolutions, Hilbert series, and degree in the regular case. It derives novel relations among Schur polynomials and connects them to the Skolem-Mahler-Lech theorem, offering geometric insights into the structure of integer zeros of exponential polynomials.

ABSTRACT

Motivated by the famous Skolem-Mahler-Lech theorem we initiate in this paper the study of a natural class of determinantal varieties which we call {\em Vandermonde varieties}. They are closely related to the varieties consisting of all linear recurrence relations of a given order possessing a non-trivial solution vanishing at a given set of integers. In the regular case, i.e., when the dimension of a Vandermonde variety is the expected one, we present its free resolution, obtain its degree and the Hilbert series. Some interesting relations among Schur polynomials are derived. Many open problems and conjectures are posed.

Motivation & Objective

• To study the algebraic and geometric structure of linear recurrence relations that vanish at a given finite set of integers.
• To define and analyze Vandermonde varieties as determinantal varieties arising from generalized Vandermonde matrices associated with recurrence roots.
• To establish the free resolution, degree, and Hilbert series of these varieties in the regular case.
• To uncover new algebraic relations among Schur polynomials through the geometry of these varieties.
• To address open problems related to the existence of arithmetic progressions in zero sets of linear recurrences, inspired by the Skolem-Mahler-Lech theorem.

Proposed method

• Define the open and closed linear recurrence varieties $V_{k;I}$ and $ar{V}_{k;I}$ as the sets of recurrence relations of order $k$ with non-trivial solutions vanishing at a given integer set $I$.
• Construct generalized Vandermonde matrices $M_{k;I}$ with entries $x_j^{i_ u}$ for $x_j$ being characteristic roots and $i_ u o I$.
• Use the vanishing of the determinant of $M_{k;I}$ to define the algebraic variety structure of $V_{k;I}$, linking it to the existence of non-trivial solutions.
• Apply techniques from commutative algebra and algebraic geometry to compute the free resolution, Hilbert series, and degree of the regular Vandermonde varieties.
• Derive identities among Schur polynomials by analyzing the structure of the ideals defining these varieties.
• Use computer-assisted search to explore cases of small $k$ and $m$, testing dimensionality, emptiness, and presence of arithmetic progressions in zero sets.

Experimental results

Research questions

• RQ1 For which pairs $(k;I)$ is the variety $V_{k;I}$ non-empty, and what is its dimension?
• RQ2 Under what conditions must a solution of a linear recurrence vanishing at $I$ vanish infinitely often, particularly forming an arithmetic progression?
• RQ3 Are there only finitely many exceptions to the rule that if $I$ consists of two arithmetic progressions with the same difference, then any solution vanishing on $I$ must have an infinite arithmetic progression of zeros?
• RQ4 Can the maximal number of integer zeros of a non-degenerate linear recurrence of order $k$ be bounded, and what is the structure of such zero sets?
• RQ5 What is the relationship between the geometry of Vandermonde varieties and the existence of additional integer zeros or arithmetic progressions in recurrence solutions?

Key findings

• The free resolution, degree, and Hilbert series of regular Vandermonde varieties are explicitly computed, providing a complete algebraic description in the expected dimension case.
• The variety $V_{k;I}$ is empty only for specific $4$-tuples such as $(0,1,3,7)$ and $(0,1,3,9)$, and their duals, suggesting these may be the only exceptions.
• For $k=3$, $m=4$, the pairs $(0,1,4,6)$ and $(0,1,4,13)$ force solutions to vanish at additional points forming a $6$-tuple, indicating a mechanism for generating arithmetic progressions.
• When two differences in a $4$-tuple $I$ coincide, solutions must vanish infinitely often, though this condition is only sufficient, not necessary.
• The known lower bound for the maximal number of integer zeros $ u_k$ of a non-degenerate recurrence of order $k$ is $inom{k+1}{2} - 1$, while the upper bound is double exponential.
• For real zeros of exponential polynomials with distinct real parts of exponents, the number of real zeros is finite, suggesting a parallel question about uniform upper bounds in terms of the number of terms.