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22
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 23 (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 is of particular interest. We show that the simplest cases are equivalent to binomial systems while in general the solution structure is highly determined by the properties of the two graphs. First we recall results by Feinberg and give rigorous proofs. Secondly, we explain how the graphs determine the Newton polytopes of the system of sparse polynomials and thus determine the solution structure. The results on positive solutions from real algebraic geometry are applied to this particular situation. Examples illustrate the theoretical results.
Factoring into Coprimes in Essentially Linear Time
"... . Let S be a nite set of positive integers. A \coprime base for S" means a set P of positive integers such that (1) each element of P is coprime to every other element of P and (2) each element of S is a product of powers of elements of P . There is a natural coprime base for S. This paper int ..."
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Cited by 19 (2 self)
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. Let S be a nite set of positive integers. A \coprime base for S" means a set P of positive integers such that (1) each element of P is coprime to every other element of P and (2) each element of S is a product of powers of elements of P . There is a natural coprime base for S. This paper introduces an algorithm that computes the natural coprime base for S in essentially linear time. The best previous result was a quadratictime algorithm of Bach, Driscoll, and Shallit. This paper also shows how to factor S into elements of P in essentially linear time. The algorithms apply to any free commutative monoid with fast algorithms for multiplication, division, and greatest common divisors; e.g., monic polynomials over a eld. They can be used as a substitute for prime factorization in many applications. 1.
An O(n³) Algorithm for Frobenius Normal Form
 IN PROCEEDINGS OF THE 1998 INTERNATIONAL SYMPOSIUM ON SYMBOLIC AND ALGEBRAIC COMPUTATION
, 1997
"... We describe an O(n³) field operations algorithm for computing the Frobenius normal form of an n \Theta n matrix. As applications we get O(n³) algorithms for two other classical problems: computing the minimal polynomial of a matrix and testing two matrices for similarity. Assuming standard matrix mu ..."
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Cited by 15 (2 self)
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We describe an O(n³) field operations algorithm for computing the Frobenius normal form of an n \Theta n matrix. As applications we get O(n³) algorithms for two other classical problems: computing the minimal polynomial of a matrix and testing two matrices for similarity. Assuming standard matrix multiplication, the previously best known deterministic complexity bound for all three problems is O(n^4).
A Linear Space Algorithm for Computing the Hermite Normal Form
 Proceedings ISSAC 2001, Lecture Notes in Computer Sci., 2146
, 2001
"... Computing the Hermite Normal Form of an n n integer matrix using the best current algorithms typically requires O(n 3 log M) space, where M is a bound on the entries of the input matrix. Although polynomial in the input size (which is O(n 2 log M)), this space blowup can easily become a seriou ..."
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Cited by 14 (2 self)
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Computing the Hermite Normal Form of an n n integer matrix using the best current algorithms typically requires O(n 3 log M) space, where M is a bound on the entries of the input matrix. Although polynomial in the input size (which is O(n 2 log M)), this space blowup can easily become a serious issue in practice when working on big integer matrices. In this paper we present a new algorithm for computing the Hermite Normal Form which uses only O(n 2 log M) space (i.e., essentially the same as the input size). When implemented using standard algorithms for integer and matrix multiplication, our algorithm has the same time complexity of the asymptotically fastest (but space inecient) algorithms. We also present a heuristic algorithm for HNF that achieves a substantial speedup when run on randomly generated input matrices.
Extremal real algebraic geometry and Adiscriminants
, 2006
"... We present a new, far simpler family of counterexamples to Kushnirenko’s Conjecture. Along the way, we illustrate a computerassisted approach to finding sparse polynomial systems with maximally many real roots, thus shedding light on the nature of optimal upper bounds in real fewnomial theory. We ..."
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Cited by 13 (6 self)
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We present a new, far simpler family of counterexamples to Kushnirenko’s Conjecture. Along the way, we illustrate a computerassisted approach to finding sparse polynomial systems with maximally many real roots, thus shedding light on the nature of optimal upper bounds in real fewnomial theory. We use a powerful recent formula for the Adiscriminant, and give new bounds on the topology of certain Adiscriminant varieties. A consequence of the latter result is a new upper bound on the number of topological types of certain real algebraic sets defined by sparse polynomial equations.
Computing invariants of simplicial manifolds
, 2004
"... Abstract. This is a survey of known algorithms in algebraic topology with a focus on finite simplicial complexes and, in particular, simplicial manifolds. Wherever possible an elementary approach is chosen. This way the text may also serve as a condensed but very basic introduction to the algebraic ..."
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Cited by 8 (3 self)
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Abstract. This is a survey of known algorithms in algebraic topology with a focus on finite simplicial complexes and, in particular, simplicial manifolds. Wherever possible an elementary approach is chosen. This way the text may also serve as a condensed but very basic introduction to the algebraic topology of simplicial manifolds. This text will appear as a chapter in the forthcoming book “Triangulated Manifolds with Few Vertices ” by Frank H. Lutz. The purpose of this chapter is to survey what is known about algorithms for the computation of algebraic invariants of topological spaces. Primarily, we use finite simplicial complexes as our model of topological spaces; for a discussion of different views see Section 4. On the way we give explicit definitions or constructions of all invariants presented. Note that we did not try to phrase all the results in their greatest generality. Similarly, we focus on invariants for which actual implementations exist. The reader is referred to Bredon’s monograph [2] for the wider perspective. For a related survey see Vegter [44]. 1. Homology
A Solution to the Extended GCD Problem with Applications
 IN PROC. INT'L. SYMP. ON SYMBOLIC AND ALGEBRAIC COMPUTATION: ISSAC '97
, 1997
"... This paper considers a variation of the extended gcd problem: the "modulo N extended gcd problem". Given an integer row vector [a i ] n i=1 , the modulo N extended gcd problem asks for an integer vector [c i ] n i=1 such that gcd( n X i=1 c i a i ; N) = gcd(a1 ; a2 ; : : : ; an ; N): ..."
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Cited by 7 (4 self)
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This paper considers a variation of the extended gcd problem: the "modulo N extended gcd problem". Given an integer row vector [a i ] n i=1 , the modulo N extended gcd problem asks for an integer vector [c i ] n i=1 such that gcd( n X i=1 c i a i ; N) = gcd(a1 ; a2 ; : : : ; an ; N): A deterministic algorithm is presented which returns an exceptionally small solution for a given instance of the problem: both max n i=1 jc i j and the number of nonzero c i 's will be bounded by O(log N ). The gcd algorithm presented here has numerous applications and has already led to faster algorithms for computing row reduced echelon forms of integer matrices and solving systems of linear Diophantine equations. In this paper we show how to apply our gcd algorithm to the problem of computing small pre and postmultipliers for the Smith normal of an integer matrix.
Fast Computation of Hermite Normal Forms of Random Integer Matrices
"... This paper is about how to compute the Hermite normal form of a random integer matrix in practice. We propose significant improvements to the algorithm by Micciancio and Warinschi, and extend these techniques to the computation of the saturation of a matrix. Tables of timings confirm the efficiency ..."
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Cited by 7 (0 self)
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This paper is about how to compute the Hermite normal form of a random integer matrix in practice. We propose significant improvements to the algorithm by Micciancio and Warinschi, and extend these techniques to the computation of the saturation of a matrix. Tables of timings confirm the efficiency of this approach. To our knowledge, our implementation is the fastest implementation for computing Hermite normal form for large matrices with large entries. Key words: Hermite normal form, exact linear algebra 1.
Computational topology
 Algorithms and Theory of Computation Handbook
, 2010
"... According to the Oxford English Dictionary, the word topology is derived of topos ( � ) meaning place, andlogy ( ���), a variant of the verb ´��� � , meaning to speak. As such, topology speaks about places: how local neighborhoods connect to each other to form a space. Computational topology, in t ..."
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Cited by 5 (3 self)
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According to the Oxford English Dictionary, the word topology is derived of topos ( � ) meaning place, andlogy ( ���), a variant of the verb ´��� � , meaning to speak. As such, topology speaks about places: how local neighborhoods connect to each other to form a space. Computational topology, in turn, undertakes the challenge of studying topology using a computer.
First Steps in Algorithmic Real Fewnomial Theory
, 2008
"... Fewnomial theory began with explicit bounds — solely in terms of the number of variables and monomial terms — on the number of real roots of systems of polynomial equations. Here we take the next logical step of investigating the corresponding existence problem: Let FEASR denote the problem of decid ..."
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Cited by 5 (5 self)
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Fewnomial theory began with explicit bounds — solely in terms of the number of variables and monomial terms — on the number of real roots of systems of polynomial equations. Here we take the next logical step of investigating the corresponding existence problem: Let FEASR denote the problem of deciding whether a given system of multivariate polynomial equations with integer coefficients has a real root or not. We describe a phasetransition for when m is large enough to make FEASR be NPhard, when restricted to inputs consisting of a single nvariate polynomial with exactly m monomial terms: polynomialtime for m≤n + 2 (for any fixed n) and NPhardness for m ≥ n + n ε (for n varying and any fixed ε> 0). Because of important connections between FEASR and Adiscriminants, we then study some new families of Adiscriminants whose signs can be decided within polynomialtime. (Adiscriminants contain all known resultants as special cases, and the latter objects are central in algorithmic algebraic geometry.) Baker’s Theorem from diophantine approximation arises as a key tool. Along the way, we also derive new quantitative bounds on the real zero