On generalizing Descartes' rule of signs to hypersurfaces

Elisenda Feliu*, Máté L. Telek

*Corresponding author for this work

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2 Citations (Scopus)
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Abstract

We give partial generalizations of the classical Descartes' rule of signs to multivariate polynomials (with real exponents), in the sense that we provide upper bounds on the number of connected components of the complement of a hypersurface in the positive orthant. In particular, we give conditions based on the geometrical configuration of the exponents and the sign of the coefficients that guarantee that the number of connected components where the polynomial attains a negative value is at most one or two. Our results fully cover the cases where such an upper bound provided by the univariate Descartes' rule of signs is one. This approach opens a new route to generalize Descartes' rule of signs to the multivariate case, differing from previous works that aim at counting the number of positive solutions of a system of multivariate polynomial equations.

Original languageEnglish
Article number108582
JournalAdvances in Mathematics
Volume408
ISSN0001-8708
DOIs
Publication statusPublished - 29 Oct 2022

Bibliographical note

Publisher Copyright:
© 2022 The Author(s)

Keywords

  • Connectivity
  • Convex function
  • Newton polytope
  • Semi-algebraic set
  • Signomial

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