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Embedding polynomial systems into vertically parametrised families: A case study on ODEbase

Oliver Daisey, Yue Ren, Yuvraj Singh

TL;DR

The work tackles embedding a fixed polynomial system $F$ into vertically parametrised families $\mathcal{F}$ to preserve generic algebraic properties such as dimension and root counts. It develops empirical criteria based on Macaulay-matrix minors and a greedy monomial-translation algorithm to identify and construct good embeddings, and it analyzes the inherent difficulty of the alignment problem. Across mass-action ODEbase instances, a simple score that minimizes monomial diversity correlates with desirable embeddings, and the proposed greedy alignment performs well in practice, aided by a public OSCAR-based interface. This yields practical methods for understanding parametrised systems and enables efficient access to most ODEbase models through OscarODEbase.jl, with potential impact on tropical-geometric analyses of steady-state varieties.

Abstract

Vertically parametrised polynomial systems are a particular nice class of parametrised polynomial systems for which a lot of interesting algebraic information is encoded in its combinatorics. Given a fixed polynomial system, we empirically study what constitutes a good vertically parametrised polynomial system that gives rise to it and how to construct said vertically parametrised polynomial system. For data, we use all polynomial systems in ODEbase, which we have transcribed to an OSCAR readable format, and made available as a Julia package OscarODEbase.

Embedding polynomial systems into vertically parametrised families: A case study on ODEbase

TL;DR

The work tackles embedding a fixed polynomial system into vertically parametrised families to preserve generic algebraic properties such as dimension and root counts. It develops empirical criteria based on Macaulay-matrix minors and a greedy monomial-translation algorithm to identify and construct good embeddings, and it analyzes the inherent difficulty of the alignment problem. Across mass-action ODEbase instances, a simple score that minimizes monomial diversity correlates with desirable embeddings, and the proposed greedy alignment performs well in practice, aided by a public OSCAR-based interface. This yields practical methods for understanding parametrised systems and enables efficient access to most ODEbase models through OscarODEbase.jl, with potential impact on tropical-geometric analyses of steady-state varieties.

Abstract

Vertically parametrised polynomial systems are a particular nice class of parametrised polynomial systems for which a lot of interesting algebraic information is encoded in its combinatorics. Given a fixed polynomial system, we empirically study what constitutes a good vertically parametrised polynomial system that gives rise to it and how to construct said vertically parametrised polynomial system. For data, we use all polynomial systems in ODEbase, which we have transcribed to an OSCAR readable format, and made available as a Julia package OscarODEbase.
Paper Structure (10 sections, 3 theorems, 21 equations, 3 figures, 1 algorithm)

This paper contains 10 sections, 3 theorems, 21 equations, 3 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Background
  3. Parametrised polynomial systems
  4. Vertically parametrised ideals
  5. Macaulay matrices
  6. Finding criteria for good embeddings
  7. Experiment setup
  8. Experimental results
  9. Finding good embeddings
  10. OscarODEbase.jl

Key Result

Lemma 2.7

Let $I\subseteq K[x^\pm]$ be a complete intersection of codimension $r$, and let $F=\{f_1,\dots,f_r\}$ be any generating set of $I$. Let $\mathcal{I}_F\subseteq K[a][x^\pm]$ be the vertically parametrised ideal induced by $F$, and let $\mathcal{I}_{F,K(a)}$ denote the generic specialisation. Then $\

Figures (3)

  • Figure 1: The tropical intersection products of one tropicalized binomial ideal (blue) and its intersection product with three different tropical lines (red).
  • Figure 2: The optimal alignment of $S_1, S_2, S_3$, and an unoptimal alignment obtained by greedily aligning $S_1$ and $S_2$ first.
  • Figure 3: Results of \ref{['alg:greedyAlignment']} in \ref{['ex:greedyAlignment']}. Black points are the original and optimal score, white points are average of 10 runs, and the bars indicate the spread of scores. Black points replaced by asteriks indicate systems whose scores have been scaled by a constant factor to fit the figure.

Theorems & Definitions (18)

  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Example 2.6
  • Lemma 2.7
  • proof
  • Lemma 2.8
  • proof
  • Definition 2.9
  • ...and 8 more