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 journal › Journal article › Research › peer-review
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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