[Paper Review] On the Existence of Semi-Regular Sequences

Timothy J. Hodges, Molina, S. D.+1 more|arXiv (Cornell University)|Dec 25, 2014

Polynomial and algebraic computation5 references3 citations

TL;DR

This paper establishes fundamental existence and non-existence conditions for semi-regular sequences over $\mathbb{F}_2$ in the algebra $B^{(n)} = \mathbb{F}_2[X_1,\dots,X_n]/(X_1^2,\dots,X_n^2)$. It proves that semi-regular elements of degree $d$ can only exist when $n \leq 3d$, and shows this bound is sharp for infinitely many $n$ when $d = 2^t$ and $n = 3d$. It further demonstrates that for any fixed degree vector $\mathbf{d} = (d_1,\dots,d_m)$, semi-regular sequences of length $m$ do not exist for sufficiently large $n$, resolving a key open question on the genericity of such sequences.

ABSTRACT

Semi-regular sequences over $\mathbb{F}_2$ are sequences of homogeneous elements of the algebra $ B^{(n)}=\mathbb{F}_2[X_1,...,X_n]/(X_1^2,...,X_n^2) $, which have as few relations between them as possible. They were introduced in order to assess the complexity of Gröbner basis algorithms such as ${\bf F}_4, {\bf F}_5$ for the solution of polynomial equations. Despite the experimental evidence that semi-regular sequences are common, it was unknown whether there existed semi-regular sequences for all $n$, except in extremely trivial situations. We prove some results on the existence and non-existence of semi-regular sequences. In particular, we show that if an element of degree $d$ in $B^{(n)}$ is semi-regular, then we must have $n\leq 3d$. Also, we show that if $d=2^t$ and $n=3d$ there exits a semi-regular element of degree $d$ establishing that the bound is sharp for infinitely many $n$. Finally, we generalize the result of non-existence of semi-regular elements to the case of sequences of a fixed length $m$.

Motivation & Objective

• To resolve the long-standing open question of whether semi-regular sequences exist for all $n$ in $B^{(n)}$.
• To determine the precise conditions under which homogeneous elements of degree $d$ in $B^{(n)}$ are semi-regular.
• To generalize the non-existence result from individual elements to sequences of fixed length $m$ and fixed degree vector $\mathbf{d}$.
• To clarify the sharpness of the $n \leq 3d$ bound via explicit constructions using symmetric polynomials.
• To investigate the asymptotic behavior of semi-regular sequences and their density in polynomial rings over $\mathbb{F}_2$.

Proposed method

• Uses the Hilbert series and index of a graded ring to characterize semi-regularity, defining $\operatorname{Ind}(B/I)$ as the first degree where $I$ captures the entire homogeneous component.
• Applies the criterion that a sequence $\lambda_1,\dots,\lambda_m$ is semi-regular iff $HS_{(\lambda_1,\dots,\lambda_m)}(z) = \left[ \frac{(1+z)^n}{\prod_{i=1}^m (1+z^{d_i})} \right]$, where the bracket denotes truncation at the index of the series.
• Analyzes the positivity of coefficients in the expansion of $\frac{(1+z)^n}{1+z^d}$ to determine the index and derive bounds on $\operatorname{Ind}\left(\frac{(1+z)^n}{1+z^d}\right)$.
• Employs combinatorial estimates involving binomial coefficients $\binom{2n}{k}$ and their differences to show that $\gamma(2n,k,d) > 0$ for large $n$, implying $\operatorname{Ind} > \frac{n}{2} + d$.
• Uses the function $\tau_{\mathbf{d}}(n)$, representing the expected index of the Hilbert series, to derive a linear lower bound $\tau_{\mathbf{d}}(n) \geq rn + c$ with $r > \frac{1}{2}$, which is key to the non-existence proof.
• Combines Theorem 4.7 on the failure degree $D_{\mathrm{ff}}(\lambda_j)$ with the index condition to derive a contradiction if a semi-regular sequence exists for large $n$, proving non-existence.

Experimental results

Research questions

• RQ1 For which values of $n$ and $d$ does there exist a semi-regular element of degree $d$ in $B^{(n)}$?
• RQ2 Is the bound $n \leq 3d$ for semi-regular elements sharp, and for which $d$ and $n$ is it achieved?
• RQ3 Can semi-regular sequences of fixed length $m$ and fixed degree vector $\mathbf{d} = (d_1,\dots,d_m)$ exist for arbitrarily large $n$?
• RQ4 What is the asymptotic behavior of the proportion of semi-regular sequences as $n \to \infty$?
• RQ5 Are there sporadic values of $(n,m)$ where the proportion of semi-regular sequences is unexpectedly low, and what causes this?

Key findings

• A homogeneous element of degree $d \geq 2$ in $B^{(n)}$ can only be semi-regular if $n \leq 3d$, establishing a necessary upper bound on $n$ for existence.
• The bound $n \leq 3d$ is sharp: when $d = 2^t$ and $n = 3d$, the symmetric polynomial $\sigma_{d,n} = \sum_{1 \leq i_1 < \cdots < i_d \leq n} x_{i_1} \cdots x_{i_d}$ is semi-regular, proving the bound is tight for infinitely many $n$.
• For any fixed degree vector $\mathbf{d} = (d_1,\dots,d_m)$ with $d_j \geq 2$ for some $j$, there exists an $N$ such that no semi-regular sequence of type $\mathbf{d}$ exists in $B^{(n)}$ for all $n \geq N$, generalizing the non-existence result beyond single elements.
• The proportion of semi-regular sequences of homogeneous elements in $n$ variables tends to one as $n \to \infty$, suggesting genericity in the limit, though not uniform across all $m(n)$.
• Low proportions of semi-regular sequences occur at sporadic $n,m$ pairs like $(10,12)$, $(11,15)$, and $(15,14)$, corresponding to zero coefficients in the Hilbert series at the index, indicating potential exceptions to density conjectures.
• The failure degree $D_{\mathrm{ff}}(\lambda_j)$ is strictly less than the index $\operatorname{Ind}(\lambda_1,\dots,\lambda_m)$ for large $n$, contradicting semi-regularity and proving non-existence via contradiction.