# Estimates for the coefficients of differential dimension polynomials

## TL;DR

This paper provides computable bounds for the coefficients of differential dimension polynomials of algebraic-differential systems, answering a long-standing question and improving bounds for components of specified differential type.

## Contribution

It offers the first computable bounds for all coefficients of Kolchin polynomials of prime components, with improved bounds for components of fixed differential type using new combinatorial results.

## Key findings

- Computed bounds for all coefficients of Kolchin polynomials.
- Established better bounds for components of fixed differential type.
- Applied combinatorial and classical theorems to improve growth estimates.

## Abstract

We answer the following long-standing question of Kolchin: given a system of algebraic-differential equations $\Sigma(x_1,\dots,x_n)=0$ in $m$ derivatives over a differential field of characteristic zero, is there a computable bound, that only depends on the order of the system (and on the fixed data $m$ and $n$), for the typical differential dimension of any prime component of $\Sigma$? We give a positive answer in a strong form; that is, we compute a (lower and upper) bound for all the coefficients of the Kolchin polynomial of every such prime component. We then show that, if we look at those components of a specified differential type, we can compute a significantly better bound for the typical differential dimension. This latter improvement comes from new combinatorial results on characteristic sets, in combination with the classical theorems of Macaulay and Gotzmann on the growth of Hilbert-Samuel functions.