Counting Real Connected Components of Trinomial Curve Intersections and m -nomial Hypersurfaces

Li, Tien-Yien; Rojas, J. Maurice; Wang, Xiaoshen

doi:10.1007/s00454-003-2834-8

Cited by 43 publications

(63 citation statements)

References 30 publications

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“…It is widely believed that significantly smaller bounds should hold. For example, Li, Rojas, and Wang [7] showed that two trinomials in 2 variables have at most 20 common solutions, which is much less than the Khovanskii bound of 20736. While significantly lower bounds are expected, we know of no reasonable conjectures about the nature of hypothetical lower bounds.…”

Section: Introductionmentioning

confidence: 99%

Polynomial systems with few real zeroes

2006

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We study some systems of polynomials whose support lies in the convex hull of a circuit, giving a sharp upper bound for their numbers of real solutions. This upper bound is non-trivial in that it is smaller than either the Kouchnirenko or the Khovanskii bounds for these systems. When the support is exactly a circuit whose affine span is Zn, this bound is 2n +1, while the Khovanskii bound is exponential in n2. The bound 2n + 1 can be attained only for non-degenerate circuits. Our methods involve a mixture of combinatorics, geometry, and arithmetic

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Section: Introductionmentioning

confidence: 99%

Polynomial systems with few real zeroes

2006

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“…This paper is devoted to the study of this problem, both in particular cases and in the general one. The results presented here are inspired by a paper by Li et al [6]. Definition 3.…”

Section: Definitionmentioning

confidence: 84%

Some Bounds for the Number of Components of Real Zero Sets of Sparse Polynomials

Perrucci

2005

Discrete Comput Geom

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We prove that the zero set of a 4-nomial in n variables in the positive orthant has at most three connected components. This bound, which does not depend on the degree of the polynomial, not only improves the best previously known bound (which was 10) but is optimal as well. In the general case we prove that the number of connected components of the zero set of an m-nomial in n variables in the positive orthant is lower than or equal to (n + 1) m−1 2 1+(m−1)(m−2)/2 , improving slightly the known bounds. Finally, we show that for generic exponents, the number of non-compact connected components of the zero set of a 5-nomial in three variables in the positive octant is at most 12. This strongly improves the best previously known bound, which was 10,384. All the bounds obtained in this paper continue to hold for real exponents.

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“…Indeed, since this is a system of two trinomials in two variables (see [13]). The key component of the construction is to find a system (4.4) that has five positive solutions.…”

Section: Proposition 42 Assume That All Solutions To (42) Are Non-dmentioning

confidence: 99%

Constructing polynomial systems with many positive solutions using tropical geometry

Hilany

2018

Rev Mat Complut

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The number of positive solutions to a system of two polynomials in two variables defined over the field of real numbers with a total of five distinct monomials cannot exceed 15. All previously known examples have at most 5 positive solutions. The main result of this paper is the construction of a system as above having 7 positive solutions. This is achieved using tools developed in tropical geometry. When the corresponding tropical hypersurfaces intersect transversally, one can easily estimate the positive solutions to the system using the classical combinatorial patchworking for complete intersections. We apply this generalization to construct a system as above having 6 positive solutions. We also show that this bound is sharp. Consequently, our main result is proved using non-transversal intersections of tropical curves.

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Counting Real Connected Components of Trinomial Curve Intersections and m -nomial Hypersurfaces

Cited by 43 publications

References 30 publications

Polynomial systems with few real zeroes

Polynomial systems with few real zeroes

Some Bounds for the Number of Components of Real Zero Sets of Sparse Polynomials

Constructing polynomial systems with many positive solutions using tropical geometry

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