Dima Grigoriev

Title: Dima Grigoriev

Research Question: How can we estimate the leading coefficient of the Hilbert-Kolchin polynomial for a left Lm-module with a given differential type?

Methodology: The research uses techniques from algebra, particularly the Weil algebra and linear partial differential equations. It focuses on systems of linear partial differential equations and their solutions. The study employs the concept of the Bernstein filtration to analyze the complexity of solving linear systems over algebras of fractions.

Results: The main result is a bound on the leading coefficient of the Hilbert-Kolchin polynomial, expressed as a function of the differential type and the order of the differential operators. This bound is a weak analogue of the B´ezout inequality for differential modules. The research also provides complexity bounds on quasi-inverse matrices over the algebra of fractions and presents a theorem on the solvability of systems of linear equations over this algebra.

Implications: The results of this research have implications for the field of algebra and differential equations. The bounds on the leading coefficient of the Hilbert-Kolchin polynomial provide insight into the structure of left Lm-modules with a given differential type. The complexity bounds on quasi-inverse matrices and the theorem on the solvability of systems of linear equations can be applied in other areas of mathematics and computer science.

Future Work: Further research could focus on improving the bounds on the leading coefficient of the Hilbert-Kolchin polynomial for larger values of m−t. Additionally, it would be interesting to study the solvability of systems of linear equations over algebras of fractions in other contexts and to explore the connections between this problem and other areas of mathematics.

Link to Article: https://arxiv.org/abs/0311053v1 Authors: arXiv ID: 0311053v1