Weighted-homogeneous structure of polynomial equations
FWF Standalone Project P34872

Goals

The goal of the FWF project P34872 is to investigate the impact of certain structures of polynomial equations on the complexity of their resolution. More precisely, given a structure, we are interested in three complementary questions:

  • identify properties of systems of differential equations which make them well-behaved in some sense;
  • prove genericity (how common are those properties?) and complexity (how fast can we hope to make the algorithms?) results under those hypotheses;
  • design algorithms for solving the systems of equations, dedicated to the structure, and controlled by those complexity bounds.

This approach has in the past led to results on a wide class of structures, including homogeneous systems and their weighted generalization, multi-homogeneous systems, determinantal systems, sparse systems…

This project aims at studying two generalizations of the existing study of weighted-homogeneous systems:

  • systems which are pluri-weighted homogeneous, namely weighted-homogeneous for several systems of weights;
  • systems which are weighted-homogeneous for systems of weights including null or negative weights.

Results

  • Short proofs of ideal membership
    C. Hofstadler, T. Verron
    Journal of Symbolic Computation, Volume 125, 2024
    LinksLinksLiens: .pdf, arXiv, doi:10.1016/j.jsc.2024.102325, .bib
  • On the computation of Gröbner bases for matrix-weighted homogeneous systems
    T. Verron
    Journal of Symbolic Computation, Volume 125, 2024
    LinksLinksLiens: .pdf, arXiv, doi:10.1016/j.jsc.2024.102327, .bib
  • Signature Gröbner bases in free algebras over rings
    C. Hofstadler, T. Verron
    International Symposium on Symbolic and Algebraic Computation (ISSAC) 2023
    LinksLinksLiens: .pdf, arXiv, doi:10.1145/3597066.3597071
  • Universal Analytic Gröbner Bases and Tropical Geometry
    T. Vaccon, T. Verron
    International Symposium on Symbolic and Algebraic Computation (ISSAC) 2023
    LinksLinksLiens: .pdf, doi:10.1145/3597066.3597110
  • Transcendence certificates of D-finite functions
    M. Kauers, C. Koutschan, T. Verron
    International Symposium on Symbolic and Algebraic Computation (ISSAC) 2023
    LinksLinksLiens: .pdf, arXiv, doi:10.1145/3597066.3597091
  • On Polynomial Ideals and Overconvergence in Tate Algebras
    X. Caruso, T. Vaccon, T. Verron
    International Symposium on Symbolic and Algebraic Computation (ISSAC) 2022
    LinksLinksLiens: .pdf, arXiv, doi:10.1145/3476446.3535491, .bib