2023/08/23 by Claude Quitté, Quitté, Claude, Claire Tête +1
Mathematics · #13C15 #13D02 #13D40 #13P15 #2020: 13C05 #Advanced Differential Equations and Dynamical Systems #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics #Mathematics and Applications
paper · pdf · doi:10.48550/arxiv.2308.12232
openalex publication_date 2023/08/23 · openalex created_date 2023/08/25 · openalex updated_date 2026/07/28
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.