## Abstract

We answer the following long-standing question of Kolchin: given

a system of algebraic-dierential equations (x1; : : : ; xn) = 0 in m derivatives

over a dierential eld of characteristic zero, is there a computable bound,

that only depends on the order of the system (and on the xed data m and n),

for the typical dierential dimension of any prime component of ? We give

a positive answer in a strong form; that is, we compute a (lower and upper)

bound for all the coecients of the Kolchin polynomial of every such prime

component. We then show that, if we look at those components of a specied

dierential type, we can compute a signicantly better bound for the typical

dierential 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.

a system of algebraic-dierential equations (x1; : : : ; xn) = 0 in m derivatives

over a dierential eld of characteristic zero, is there a computable bound,

that only depends on the order of the system (and on the xed data m and n),

for the typical dierential dimension of any prime component of ? We give

a positive answer in a strong form; that is, we compute a (lower and upper)

bound for all the coecients of the Kolchin polynomial of every such prime

component. We then show that, if we look at those components of a specied

dierential type, we can compute a signicantly better bound for the typical

dierential 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.

Original language | English |
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Pages (from-to) | 2959-2985 |

Journal | Mathematics of Computation |

Volume | 88 |

Issue number | 0 |

Early online date | 28 Mar 2019 |

DOIs | |

Publication status | Published - 2019 |