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Bounds for D-Algebraic Closure Properties

Manuel Kauers, Raphael Pages

TL;DR

This work extends the study of closure properties for $D$-algebraic functions by deriving explicit bounds on the total degree of the polynomial differential equations that define composed, added, multiplied, divided, and otherwise combined $D$-algebraic functions. While order bounds are readily obtained from transcendence-degree arguments, the core contribution is a general degree bound under a genericity (

Abstract

We provide bounds on the size of polynomial differential equations obtained by executing closure properties for D-algebraic functions. While it is easy to obtain bounds on the order of these equations, it requires some more work to derive bounds on their degree. Here we give bounds that apply under some technical condition about the defining differential equations.

Bounds for D-Algebraic Closure Properties

TL;DR

This work extends the study of closure properties for -algebraic functions by deriving explicit bounds on the total degree of the polynomial differential equations that define composed, added, multiplied, divided, and otherwise combined -algebraic functions. While order bounds are readily obtained from transcendence-degree arguments, the core contribution is a general degree bound under a genericity (

Abstract

We provide bounds on the size of polynomial differential equations obtained by executing closure properties for D-algebraic functions. While it is easy to obtain bounds on the order of these equations, it requires some more work to derive bounds on their degree. Here we give bounds that apply under some technical condition about the defining differential equations.
Paper Structure (7 sections, 11 theorems, 38 equations, 1 figure)

This paper contains 7 sections, 11 theorems, 38 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Dimensions
  3. Setting
  4. Degree bounds in complete intersections
  5. Corollaries on degree bounds for algebraic operations
  6. Bounds for the composition of D-algebraic functions
  7. Variable elimination in special cases

Key Result

proposition 1

CoLiOS07 For every homogeneous ideal $I$ of $R$ there exists a polynomial $HP_I\in\mathbb Q[t]$ such that $HP_I(i)=HF_I(i)$ for all sufficiently large $i\in\mathbb N$.

Figures (1)

  • Figure 1: order-number of monomials curves

Theorems & Definitions (29)

  • definition 1
  • proposition 1
  • definition 2
  • definition 3
  • proposition 2
  • proof
  • definition 4
  • proposition 3
  • proof
  • proposition 4
  • ...and 19 more