On generalizing Descartes' rule of signs to hypersurfaces

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On generalizing Descartes' rule of signs to hypersurfaces. / Feliu, Elisenda; Telek, Máté L.

In: Advances in Mathematics, Vol. 408, 108582, 29.10.2022.

Research output: Contribution to journalJournal articlepeer-review

Harvard

Feliu, E & Telek, ML 2022, 'On generalizing Descartes' rule of signs to hypersurfaces', Advances in Mathematics, vol. 408, 108582. https://doi.org/10.1016/j.aim.2022.108582

APA

Feliu, E., & Telek, M. L. (2022). On generalizing Descartes' rule of signs to hypersurfaces. Advances in Mathematics, 408, [108582]. https://doi.org/10.1016/j.aim.2022.108582

Vancouver

Feliu E, Telek ML. On generalizing Descartes' rule of signs to hypersurfaces. Advances in Mathematics. 2022 Oct 29;408. 108582. https://doi.org/10.1016/j.aim.2022.108582

Author

Feliu, Elisenda ; Telek, Máté L. / On generalizing Descartes' rule of signs to hypersurfaces. In: Advances in Mathematics. 2022 ; Vol. 408.

Bibtex

@article{4163003a87f24a7ca248e40a378f1766,
title = "On generalizing Descartes' rule of signs to hypersurfaces",
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.",
keywords = "Connectivity, Convex function, Newton polytope, Semi-algebraic set, Signomial",
author = "Elisenda Feliu and Telek, {M{\'a}t{\'e} L.}",
note = "Publisher Copyright: {\textcopyright} 2022 The Author(s)",
year = "2022",
month = oct,
day = "29",
doi = "10.1016/j.aim.2022.108582",
language = "English",
volume = "408",
journal = "Advances in Mathematics",
issn = "0001-8708",
publisher = "Academic Press",

}

RIS

TY - JOUR

T1 - On generalizing Descartes' rule of signs to hypersurfaces

AU - Feliu, Elisenda

AU - Telek, Máté L.

N1 - Publisher Copyright: © 2022 The Author(s)

PY - 2022/10/29

Y1 - 2022/10/29

N2 - 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.

AB - 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.

KW - Connectivity

KW - Convex function

KW - Newton polytope

KW - Semi-algebraic set

KW - Signomial

UR - http://www.scopus.com/inward/record.url?scp=85135109753&partnerID=8YFLogxK

U2 - 10.1016/j.aim.2022.108582

DO - 10.1016/j.aim.2022.108582

M3 - Journal article

AN - SCOPUS:85135109753

VL - 408

JO - Advances in Mathematics

JF - Advances in Mathematics

SN - 0001-8708

M1 - 108582

ER -

ID: 315761943