Descartes' rule of signs for arbitrary fewnomial systems
Abstract
The classical Descartes' rule of signs bounds the number of positive roots of a real polynomial in one variable by the number of sign changes of its coefficients. This bound only depends on the order of the exponents and on the signs of the coefficients, that is, on the oriented matroids of the exponent and coefficient matrices. We give a multivariate generalization of this rule to arbitrary fewnomial systems: for systems of n real polynomial equations in n variables with n+k+1 monomials, we obtain upper bounds for the number of nondegenerate positive solutions which only depend on sign conditions on the maximal minors of the coefficient matrix and of the exponent matrix and its liftings. For k=1 they are the Descartes rules of signs for circuits of Bihan-Dickenstein and Bihan-Dickenstein-Forsgård. Our approach starts from the Gale duality and the Khovanskii-Rolle argument of Bihan and Sottile, which reduce the problem to counting common zeros of iterated Jacobians in a polyhedron Δ. This polyhedron is a chamber of a hyperplane arrangement whose combinatorics is determined by the oriented matroid of the coefficient matrix. We bound the number of these zeros by the Bézout number minus the zeros forced in the other chambers by a topological degree, which we compute by a facet-count formula involving the signs of minors of a Gale dual matrix of the exponent matrix. This leads to a recursion over the flats of the arrangement, which ends on lines with the Descartes rules for circuits. In this way the coefficients enter the bounds through the combinatorics of the arrangement, and the exponents through sign conditions on minors.
Related
- Reduced Solutions of Toda Systems for Arbitrary Complex Simple Lie Algebras
- Funnel control with input filter for nonlinear systems with arbitrary relative degree
- Topological degree of fewnomial maps
- Regularity and Rivière's GL(m)-Gauge Construction for Elliptic Systems with Antisymmetric Potentials in Arbitrary Dimensions
- Descartes' Circle Theorem, Princess Elizabeth, and Spinors
BibTeX
@misc{bihan2026descartes,
title = {{Descartes' rule of signs for arbitrary fewnomial systems}},
author = {Frédéric Bihan},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.06435},
eprint = {2610.06435},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.06435},
}
RIS
TY - UNPB
TI - Descartes' rule of signs for arbitrary fewnomial systems
AU - Frédéric Bihan
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2610.06435
UR - https://arxiv.org/abs/2610.06435
AB - The classical Descartes' rule of signs bounds the number of positive roots of a real polynomial in one variable by the number of sign changes of its coefficients. This bound only depends on the order of the exponents and on the signs of the coefficients, that is, on the oriented matroids of the exponent and coefficient matrices. We give a multivariate generalization of this rule to arbitrary fewnomial systems: for systems of n real polynomial equations in n variables with n+k+1 monomials, we obtain upper bounds for the number of nondegenerate positive solutions which only depend on sign conditions on the maximal minors of the coefficient matrix and of the exponent matrix and its liftings. For k=1 they are the Descartes rules of signs for circuits of Bihan-Dickenstein and Bihan-Dickenstein-Forsgård. Our approach starts from the Gale duality and the Khovanskii-Rolle argument of Bihan and Sottile, which reduce the problem to counting common zeros of iterated Jacobians in a polyhedron Δ. This polyhedron is a chamber of a hyperplane arrangement whose combinatorics is determined by the oriented matroid of the coefficient matrix. We bound the number of these zeros by the Bézout number minus the zeros forced in the other chambers by a topological degree, which we compute by a facet-count formula involving the signs of minors of a Gale dual matrix of the exponent matrix. This leads to a recursion over the flats of the arrangement, which ends on lines with the Descartes rules for circuits. In this way the coefficients enter the bounds through the combinatorics of the arrangement, and the exponents through sign conditions on minors.
ER -