Bivariate systems of polynomial equations with roots of high multiplicity

I. Nikitin

Bivariate systems of polynomial equations with roots of high multiplicity

Algebraic Geometry 2021-10-26 v4 Combinatorics

Authors: I. Nikitin

Abstract

Given a bivariate system of polynomial equations with fixed support sets $A, B$ it is natural to ask which multiplicities its solutions can have. We prove that there exists a system with a solution of multiplicity $i$ for all $i$ in the range $\{0,1,...,|A|-|\mbox{conv}(A)\ominus B|-1\}$ , where $A\ominus B$ is the set of all integral vectors that shift B to a subset of $A$ . As an application of this result we classify all pairs $(A, B)$ such that the system supported at $(A, B)$ does not have a solution of multiplicity $3$ .

Keywords

bivariate polynomial systems solution multiplicity support sets minkowski subtraction classification

Cite

@article{arxiv.1910.12541,
  title  = {Bivariate systems of polynomial equations with roots of high multiplicity},
  author = {I. Nikitin},
  journal= {arXiv preprint arXiv:1910.12541},
  year   = {2021}
}

Comments

26 pages, 11 figures (In the version 3 the classification in the theorem 4.2 was incomplete. In version 4 this bug is fixed)

Bivariate systems of polynomial equations with roots of high multiplicity

Abstract

Keywords

Cite

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