Une approche combinatoire pour le r\'esultant multivari\'e. Le r\'esultant multivari\'e pour les enfants motiv\'es

TL;DR

This paper presents a comprehensive algebraic approach to multivariate resultants using complex structures, revealing new binomial relations and formulas for elimination theory in polynomial systems.

Contribution

It introduces a novel combinatorial method leveraging finite free resolutions, Koszul complexes, and Macaulay decompositions to analyze multivariate resultants.

Findings

Derived determinantal expressions for resultants.

Established binomial relations between scalar families.

Explored properties of the fundamental linear form at critical degree.

Abstract

We provide, in a 474 pages study, a comprehensive and self-contained treatment of Resultant Theory for a homogeneous system of polynomials with several variables (as many variables as of polynomials). In a non classical way, we use the multiplicative structure of finite free resolutions, by applying it to the complex homogeneous components of the Koszul complex of the system, and this in any degree. Moreover, these complexes have Macaulay decompositions. These three pillars, multiplicative structure, Koszul complex, Macaulay decomposition, allow to establish, surprisingly to us, remarkable binomial relations between 3 families of scalars resulting from the differentials of that complexes. These binomial relations generate several notable results, like a determinantal expression of a certain denominator, depending only on the first differential, and provide in particular formulas expressing the resultant. We have explored more deeply the case of the critical degree delta. This degree produces a fundamental linear form on the homogeneous polynomial component of degree delta, including the resultant. And this fundamental linear form has allowed us to highlight a certain number of remarkable properties. Key words: Elimination theory, homogeneous polynomial system, resultant, determinant of complexes, Cayley determinant, computational algebra, finitefree resolution, Koszul complex, Euler characteristic, Grade, Multiplicative structure, Macaulay decomposition, Fitting invariants, MacRae invariant, regular sequence.