PreprintCC BY-NC 4.0HAL
Intersection multiplicity of a sparse curve and a trinomial curve
Abstract
Let f ∈ C[x, y] be a trinomial and let g ∈ C[x, y] be a sparse polynomial with N monomials (N ≥ 1). We address the problem of bounding the local intersection multiplicity of the bivariate polynomial system f = g = 0 at an isolated root q ∈ (C * )^ 2 only in terms of N . Our main result establishes that the local multiplicity of the system at any isolated solution q ∈ (C * ) ^2 is at most (N -1)(N +2)/ 2 , independently of the degrees of f and g. Furthermore, we know that this bound is sharp for N = 3.
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BibTeX
@misc{mousseau2026intersection,
title = {{Intersection multiplicity of a sparse curve and a trinomial curve}},
author = {Jean Mousseau},
howpublished = {HAL},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.35037},
eprint = {2610.35037},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05837591v2},
note = {CC BY-NC 4.0},
}
RIS
TY - UNPB
TI - Intersection multiplicity of a sparse curve and a trinomial curve
AU - Jean Mousseau
PY - 2026
DA - 2026/10/23
PB - HAL
DO - 10.48550/arxiv.2610.35037
UR - https://hal.science/hal-05837591v2
AB - Let f ∈ C[x, y] be a trinomial and let g ∈ C[x, y] be a sparse polynomial with N monomials (N ≥ 1). We address the problem of bounding the local intersection multiplicity of the bivariate polynomial system f = g = 0 at an isolated root q ∈ (C * )^ 2 only in terms of N . Our main result establishes that the local multiplicity of the system at any isolated solution q ∈ (C * ) ^2 is at most (N -1)(N +2)/ 2 , independently of the degrees of f and g. Furthermore, we know that this bound is sharp for N = 3.
ER -