A sharp bound on the number of real intersection points of a sparse plane curve with a line

Bihan, Frédéric; Hilany, Boulos El

doi:10.1016/j.jsc.2016.12.003

Cited by 6 publications

(3 citation statements)

References 9 publications

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“…This question is open even in the case of two variables to the best of our knowledge; see [20]. Several articles give an affirmative answer in special cases, such as the intersection of a line with the zero set of a bivariate t-nomial [1,4], or the intersection of two plane curves, where one is defined by a trinomial and the other by a t-nomial [21,Cor. 16].…”

mentioning

confidence: 99%

On the Number of Real Zeros of Random Fewnomials

Bürgisser¹,

Ergür²,

Tonelli-Cueto³

2019

SIAM J. Appl. Algebra Geometry

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Consider a system f 1 (x) = 0, . . . , fn(x) = 0 of n random real polynomial equations in n variables, where each f i has a prescribed set of exponent vectors described by a set A ⊆ N n of cardinality t. Assuming that the coefficients of the f i are independent Gaussians of any variance, we prove that the expected number of zeros of the random system in the positive orthant is bounded from above by 1 2 n−1 t n .

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confidence: 99%

On the Number of Real Zeros of Random Fewnomials

Bürgisser¹,

Ergür²,

Tonelli-Cueto³

2019

SIAM J. Appl. Algebra Geometry

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“…The bound on the number of real zeros obtained by Khovanskii is exponential in the number t. It is widely conjectured that this bound is far from optimal: in fact it is conjectured [30] that for fixed n, the number of nondegenerate positive solutions of a fewnomial system with t exponent vectors is bounded by a polynomial in t. Quite surprisingly, this question is open even for n = 2! For results in special cases, we refer to [4,1,36,23,24,3]. Moreover, there is a very interesting connection to complexity theory [22,5].…”

Section: Introductionmentioning

confidence: 99%

Real zeros of mixed random fewnomial systems

Bürgisser¹

2023

Preprint

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Consider a system f 1 (x) = 0, . . . , fn(x) = 0 of n random real polynomials in n variables, where each f i has a prescribed set of exponent vectors in a set A i ⊆ Z n of cardinality t i , whose convex hull is denoted P i . Assuming that the coefficients of the f i are independent standard Gaussian, we prove that the expected number of zeros of the random system in the positive orthant is at most (2π) − n 2 V 0 (t 1 − 1) . . . (tn − 1). Here V 0 denotes the number of vertices of the Minkowski sum P 1 + . . . + Pn. We also derive a better bound in the unmixed case where all supports A i are equal, improving upon Bürgisser et al. (SIAM J. Appl. Algebra Geom. 3(4), 2019). All arguments equally work for real exponent vectors.

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“…This bound was later improved by F. Bihan and F. Sottile [7] to e 2 +3 4 2 ( k 2 ) n k , however only a handful of sharp fewnomial bounds are known (c.f. [3,5,11]). When k = 0, the system ( ) can be reduced to a system where each polynomial is a binomial, and thus has at most one non-degenerate positive solution.…”

Section: Introductionmentioning

confidence: 99%

Characterization of circuits supporting polynomial systems with the maximal number of positive solutions

Hilany¹

2016

Preprint

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A polynomial system with n equations in n variables supported on a set W ⊂ R n of n + 2 points has at most n + 1 non-degenerate positive solutions. Moreover, if this bound is reached, then W is minimally affinely dependent, in other words, it is a circuit in R n . For any positive integer number n, we determine all circuits W ⊂ R n which can support a polynomial system with n+1 non-degenerate positive solutions. Restrictions on such circuits W are obtained using Grothendieck's real dessins d'enfant, while polynomial systems with n + 1 non-degenerate positive solutions are constructed using Viro's combinatorial patchworking.

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A sharp bound on the number of real intersection points of a sparse plane curve with a line

Cited by 6 publications

References 9 publications

On the Number of Real Zeros of Random Fewnomials

On the Number of Real Zeros of Random Fewnomials

Real zeros of mixed random fewnomial systems

Characterization of circuits supporting polynomial systems with the maximal number of positive solutions

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