Fewnomial systems with many roots, and an Adelic Tau Conjecture

Phillipson, Kaitlyn; Rojas, J. Maurice

doi:10.1090/conm/605/12111

Cited by 14 publications

(17 citation statements)

References 51 publications

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“…This technique is constructive but explicit values for the coefficients of the system are not given. Explicit systems reaching this bound have been given by Kaitlyn Phillipson and J. Maurice Rojas in [10]. It turns out that the polynomial systems constructed in [1] are maximally positive: they have m(W) + 1 positive solutions and no other complex solutions.…”

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

Maximally positive polynomial systems supported on circuits

Bihan

2015

Journal of Symbolic Computation

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A real polynomial system with support W ⊂ Z n is called maximally positive if all its complex solutions are positive solutions. A support W having n + 2 elements is called a circuit. We previously showed that the number of non-degenerate positive solutions of a system supported on a circuit W ⊂ Z n is at most m(W) + 1, where m(W) ≤ n is the degeneracy index of W. We prove that if a circuit W ⊂ Z n supports a maximally positive system with the maximal number m(W) + 1 of non-degenerate positive solutions, then it is unique up to the obvious action of the group of invertible integer affine transformations of Z n . In the general case, we prove that any maximally positive system supported on a circuit can be obtained from another one having the maximal number of positive solutions by means of some elementary transformations. As a consequence, we get for each n and up to the above action a finite list of circuits W ⊂ Z n which can support maximally positive polynomial systems. We observe that the coefficients of the primitive affine relation of such circuit have absolute value 1 or 2 and make a conjecture in the general case for supports of maximally positive systems.

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Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

Maximally positive polynomial systems supported on circuits

Bihan

2015

Journal of Symbolic Computation

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“…It is widely conjectured that these bounds are far from being optimal, but very little is known. The following question is a central open problem in fewnomial theory [25]. Question 1.1.…”

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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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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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“…When r = 2 this is easy. When |W| = 4 the bound in Proposition 5.8 is 3 and is sharp, see [19]. When |W| = 5, the bound in Proposition 5.8 is 6.…”

Section: Discrete Mixed Volume and Fewnomial Bounds For Tropical Systemsmentioning

confidence: 93%

Irrational Mixed Decomposition and Sharp Fewnomial Bounds for Tropical Polynomial Systems

Bihan

2016

Discrete Comput Geom

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Abstract. Given convex polytopes P 1 , . . . , P r ⊂ R n and finite subsets W I of the Minkowsky sums P I = i∈I P i , we consider the quantityIf W I = Z n ∩ P I and P 1 , . . . , P n are lattice polytopes in R n , then N (W) is the classical mixed volume of P 1 , . . . , P n giving the number of complex solutions of a general complex polynomial system with Newton polytopes P 1 , . . . , P n . We develop a technique that we call irrational mixed decomposition which allows us to estimate N (W) under some assumptions on the family W = (W I ). In particular, we are able to show the nonnegativity of N (W) in some important cases. A special attention is paid to the family W = (W I ) defined by W I = i∈I W i , where W 1 , . . . , W r are finite subsets of P 1 , . . . , P r . The associated quantity N (W) is called discrete mixed volume of W 1 , . . . , W r . Using our irrational mixed decomposition technique, we show that for r = n the discrete mixed volume is an upper bound for the number of nondegenerate solutions of a tropical polynomial system with supports W 1 , . . . , W n ⊂ R n . We also prove that the discrete mixed volume associated with W 1 , . . . , W r is bounded from above by the Kouchnirenko number r i=1 (|W i | − 1). For r = n this number was proposed as a bound for the number of nondegenerate positive solutions of any real polynomial system with supports W 1 , . . . , W n ⊂ R n . This conjecture was disproved, but our result shows that the Kouchnirenko number is a sharp bound for the number of nondegenerate positive solutions of real polynomial systems constructed by means of the combinatorial patchworking.

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Fewnomial systems with many roots, and an Adelic Tau Conjecture

Cited by 14 publications

References 51 publications

Maximally positive polynomial systems supported on circuits

Maximally positive polynomial systems supported on circuits

On the Number of Real Zeros of Random Fewnomials

Irrational Mixed Decomposition and Sharp Fewnomial Bounds for Tropical Polynomial Systems

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