Descartes’ Rule of Signs for Polynomial Systems Supported on Circuits

Bihan, Frédéric; Dickenstein, Alicia

doi:10.1093/imrn/rnw199

Cited by 12 publications

(14 citation statements)

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“…This happens when the exponent vectors of the original polynomial system are not in convex position and a sign condition holds on the coefficient matrix. Such sign conditions are also found in recent work giving very strong bounds on positive solutions [6,9,19] and are considered to be multivariate versions of Descartes' rule of signs.…”

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confidence: 67%

Software for the Gale transform of fewnomial systems and a Descartes rule for fewnomials

et al. 2016

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confidence: 67%

Software for the Gale transform of fewnomial systems and a Descartes rule for fewnomials

et al. 2016

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“…We will give the proof of Theorem 1.5 in Section 3.3, where we restate the sign conditions on the minors of A and B in terms of oriented matroids. Based on this approach, a generalization for multivariate polynomials systems in n variables with n + 2 distinct monomials is given in [11]. This case shows the intricacy inherent in the pursuit of a full generalization of Descartes' rule to the multivariate case.…”

Section: Application To Real Algebraic Geometrymentioning

confidence: 99%

“…is infeasible, that is, the system (11) has no solution z = (x, y) ∈ R r+n . Linear inequalities arise from the sign equalities in (11). Some of these are strict inequalities and hence techniques from linear programming do not directly apply.…”

Section: Algorithmic Verification Of Sign Conditionsmentioning

confidence: 99%

Sign Conditions for Injectivity of Generalized Polynomial Maps with Applications to Chemical Reaction Networks and Real Algebraic Geometry

et al. 2015

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We give necessary and sufficient conditions in terms of sign vectors for the injectivity of families of polynomial maps with arbitrary real exponents defined on the positive orthant. Our work relates and extends existing injectivity conditions expressed in terms of Jacobian matrices and determinants. In the context of chemical reaction networks with power-law kinetics, our results can be used to preclude as well as to guarantee multiple positive steady states. In the context of real algebraic geometry, our work recognizes a prior result of Craciun, Garcia-Puente, and Sottile, together with work of two of the authors, as the first partial multivariate generalization of the classical Descartes' rule, which bounds the number of positive real roots of a univariate real polynomial in terms of the number of sign variations of its coefficients.

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“…Requiring in addition that the conic passes through α 1 We leave it for the reader to verify that, with these coefficients, the polynomia P (x) evaluated at x = α 3 is equal to H (α 1 , α 2 , α 3 ).…”

Section: Rationality Of the Cuspidal Locusmentioning

confidence: 99%

“…In this approach, one considers the family of all polynomials which can be expressed using a fixed set (the support set) A of exponent vectors α. For example, in [1] the bound on the number of positive solutions of a system of equations supported on a circuit was sharpened by considering in addition the combinatorics of A.…”

Section: Introductionmentioning

confidence: 99%