arXiv subjects

William Simmons

Publications and source records attributed to William Simmons.

4 recordsLinked to original sources

arXiv2023

Differential Elimination and Algebraic Invariants of Polynomial Dynamical Systems

Invariant sets are a key ingredient for verifying safety and other properties of cyber-physical systems that mix discrete and continuous dynamics. We adapt the elimination-theoretic Rosenfeld-Gr\"{o}bner algorithm to systematically obtain algebraic invariants of polynomial dynamical systems without using Gr\"{o}bner bases or quantifier elimination. We identify totally real varieties as an important class for efficient invariance checking.

cs.SC↗

arXiv2019

Explicit polynomial bounds on prime ideals in polynomial rings over fields

Suppose $I$ is an ideal of a polynomial ring over a field, $I\subseteq k[x_1,\ldots,x_n]$, and whenever $fg\in I$ with degree $\leq b$, then either $f\in I$ or $g\in I$. When $b$ is sufficiently large, it follows that $I$ is prime. Schmidt-Göttsch proved that "sufficiently large" can be taken to be a polynomial in the degree of generators of $I$ (with the degree of this polynomial depending on $n$). However Schmidt-Göttsch used model-theoretic methods to show this, and did not give any indication of how large the degree of this polynomial is. In this paper we obtain an explicit bound on $b$, polynomial in the degree of the generators of $I$. We also give a similar bound for detecting maximal ideals in $k[x_1,\ldots,x_n]$.

math.AC↗

arXiv2018

Proof mining and effective bounds in differential polynomial rings

Using the functional interpretation from proof theory, we analyze nonconstructive proofs of several central theorems about polynomial and differential polynomial rings. We extract effective bounds, some of which are new to the literature, from the resulting proofs. In the process we discuss the constructive content of Noetherian rings and the Nullstellensatz in both the classical and differential settings. Sufficient background is given to understand the proof-theoretic and differential-algebraic framework of the main results.

math.LO↗

arXiv2014

On linear dependence over complete differential algebraic varieties

We extend Kolchin's results on linear dependence over projective varieties in the constants, to linear dependence over arbitrary complete differential varieties. We show that in this more general setting, the notion of linear dependence still has necessary and sufficient conditions given by the vanishing of a certain system of differential polynomials equations. We also discuss some conjectural questions around completeness and the catenary problem.

math.AG↗