Results 1  10
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766,869
Random Sparse Polynomial Systems
, 2000
"... Let f :=(f 1 ; : : : ; f n ) be a sparse random polynomial system. This means that each f i has xed support (list of possibly nonzero coe cients) and each coecient has a Gaussian probability distribution of arbitrary variance. We express the expected number of roots of f inside a region U a ..."
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Cited by 5 (2 self)
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Let f :=(f 1 ; : : : ; f n ) be a sparse random polynomial system. This means that each f i has xed support (list of possibly nonzero coe cients) and each coecient has a Gaussian probability distribution of arbitrary variance. We express the expected number of roots of f inside a region U
GLOBAL RESIDUES FOR SPARSE POLYNOMIAL SYSTEMS
, 2005
"... Abstract. We consider families of sparse Laurent polynomials f1,..., fn with a finite set of common zeroes Zf in the torus T n = (C − {0}) n. The global residue assigns to every Laurent polynomial g the sum of its Grothendieck residues over Zf. We present a new symbolic algorithm for computing the g ..."
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Cited by 1 (1 self)
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Abstract. We consider families of sparse Laurent polynomials f1,..., fn with a finite set of common zeroes Zf in the torus T n = (C − {0}) n. The global residue assigns to every Laurent polynomial g the sum of its Grothendieck residues over Zf. We present a new symbolic algorithm for computing
Lower bounds for real solutions to sparse polynomial systems
 Advances in Math
, 2005
"... Abstract. We show how to construct sparse polynomial systems that have nontrivial lower bounds on their numbers of real solutions. These are unmixed systems associated to certain polytopes. For the order polytope of a poset P this lower bound is the signimbalance of P and it holds if all maximal ch ..."
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Cited by 26 (15 self)
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Abstract. We show how to construct sparse polynomial systems that have nontrivial lower bounds on their numbers of real solutions. These are unmixed systems associated to certain polytopes. For the order polytope of a poset P this lower bound is the signimbalance of P and it holds if all maximal
Counting Stable Solutions of Sparse Polynomial Systems in Chemistry
, 2000
"... The polynomial differential system modeling the behavior of a chemical reaction is given by graph theoretic structures. The concepts from toric geometry are applied to study the steady states and stable steady states. Deformed toric varieties give some insight and enable graph theoretic interpretati ..."
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Cited by 14 (2 self)
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interpretations. The importance of the circuits in the directed graph are emphasized. The counting of positive solutions of a sparse polynomial system by B. Sturmfels is generalized to the counting of stable positive solutions in case of a polynomial differential equation. The generalization is based on a method
On the Average Number of Real Roots of Certain Random Sparse Polynomial Systems
, 1996
"... We derive an explicit formula for the expected number of real roots of certain random sparse polynomial systems. We propose (and use) a probability measure which is natural in the sense that it is invariant under a natural Gaction on the space of roots, where G is a product of orthogonal groups. O ..."
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Cited by 17 (3 self)
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We derive an explicit formula for the expected number of real roots of certain random sparse polynomial systems. We propose (and use) a probability measure which is natural in the sense that it is invariant under a natural Gaction on the space of roots, where G is a product of orthogonal groups
Solving Degenerate Sparse Polynomial Systems Faster
 Journal of Symbolic Computation
, 1999
"... This paper is dedicated to my son, Victor Lorenzo. ..."
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Cited by 25 (3 self)
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This paper is dedicated to my son, Victor Lorenzo.
Symmetric Newton Polytopes for Solving Sparse Polynomial Systems
 ADV. APPL. MATH
, 1994
"... The aim of this paper is to compute all isolated solutions to symmetric polynomial systems. Recently, it has been proved that modelling the sparse structure of the system by its Newton polytopes leads to a computational breakthrough in solving the system. In this paper, it will be shown how the Lift ..."
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Cited by 18 (9 self)
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The aim of this paper is to compute all isolated solutions to symmetric polynomial systems. Recently, it has been proved that modelling the sparse structure of the system by its Newton polytopes leads to a computational breakthrough in solving the system. In this paper, it will be shown how
EXTREMAL SPARSE POLYNOMIAL SYSTEMS OVER LOCAL FIELDS
"... Consider a system F of n polynomials in n variables, with a total of n + k distinct exponent vectors, over any local field L. We discuss conjecturally tight upper and lower bounds on the maximal number of nondegenerate roots F can have over L, with all coordinates having fixed sign or fixed first ..."
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Consider a system F of n polynomials in n variables, with a total of n + k distinct exponent vectors, over any local field L. We discuss conjecturally tight upper and lower bounds on the maximal number of nondegenerate roots F can have over L, with all coordinates having fixed sign or fixed first
A Family of Sparse Polynomial Systems Arising in Chemical Reaction Systems
, 1999
"... A class of sparse polynomial systems is investigated which is dened by a weighted directed graph and a weighted bipartite graph. They arise in the model of mass action kinetics for chemical reaction systems. In this application the number of real positive solutions within a certain affine subspace i ..."
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Cited by 24 (2 self)
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A class of sparse polynomial systems is investigated which is dened by a weighted directed graph and a weighted bipartite graph. They arise in the model of mass action kinetics for chemical reaction systems. In this application the number of real positive solutions within a certain affine subspace
Results 1  10
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766,869