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18
Semidefinite Representations for Finite Varieties
 MATHEMATICAL PROGRAMMING
, 2002
"... We consider the problem of minimizing a polynomial over a semialgebraic set defined by polynomial equalities and inequalities. When the polynomial equalities have a finite number of complex solutions and define a radical ideal we can reformulate this problem as a semidefinite programming prob ..."
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Cited by 36 (6 self)
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We consider the problem of minimizing a polynomial over a semialgebraic set defined by polynomial equalities and inequalities. When the polynomial equalities have a finite number of complex solutions and define a radical ideal we can reformulate this problem as a semidefinite programming problem. This semidefinite program involves combinatorial moment matrices, which are matrices indexed by a basis of the quotient vector space R[x 1 , . . . , x n ]/I. Our arguments are elementary and extend known facts for the grid case including 0/1 and polynomial programming. They also relate to known algebraic tools for solving polynomial systems of equations with finitely many complex solutions. Semidefinite approximations can be constructed by considering truncated combinatorial moment matrices; rank conditions are given (in a grid case) that ensure that the approximation solves the original problem at optimality.
A Descartes algorithm for polynomials with bitstream coefficients
 CASC, VOLUME 3718 OF LNCS
, 2005
"... The Descartes method is an algorithm for isolating the real roots of squarefree polynomials with real coefficients. We assume that coefficients are given as (potentially infinite) bitstreams. In other words, coefficients can be approximated to any desired accuracy, but are not known exactly. We s ..."
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Cited by 31 (3 self)
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The Descartes method is an algorithm for isolating the real roots of squarefree polynomials with real coefficients. We assume that coefficients are given as (potentially infinite) bitstreams. In other words, coefficients can be approximated to any desired accuracy, but are not known exactly. We show that a variant of the Descartes algorithm can cope with bitstream coefficients. To isolate the real roots of a squarefree real polynomial q(x) = qnx n +...+q0 with root separation ρ, coefficients qn  ≥ 1 and qi  ≤ 2 τ, it needs coefficient approximations to O(n(log(1/ρ)+τ)) bits after the binary point and has an expected cost of O(n 4 (log(1/ρ)+τ) 2) bit operations.
Positive polynomials in scalar and matrix variables, the spectral theorem, and optimization
 , in vol. Structured Matrices and Dilations. A Volume Dedicated to the Memory of Tiberiu Constantinescu
"... We follow a stream of the history of positive matrices and positive functionals, as applied to algebraic sums of squares decompositions, with emphasis on the interaction between classical moment problems, function theory of one or several complex variables and modern operator theory. The second par ..."
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Cited by 13 (3 self)
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We follow a stream of the history of positive matrices and positive functionals, as applied to algebraic sums of squares decompositions, with emphasis on the interaction between classical moment problems, function theory of one or several complex variables and modern operator theory. The second part of the survey focuses on recently discovered connections between real algebraic geometry and optimization as well as polynomials in matrix variables and some control theory problems. These new applications have prompted a series of recent studies devoted to the structure of positivity and convexity in a free ∗algebra, the appropriate setting for analyzing inequalities on polynomials having matrix variables. We sketch some of these developments, add to them and comment on the rapidly growing literature.
Polynomial Systems with Few Real Zeroes
"... Abstract. We study some systems of polynomials whose support lies in the convex hull of a circuit, giving an upper bound for their numbers of real solutions which is sharp in some instances. This upper bound is nontrivial in that it is smaller than either the Kouchnirenko or the Khovanskii bounds f ..."
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Cited by 12 (10 self)
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Abstract. We study some systems of polynomials whose support lies in the convex hull of a circuit, giving an upper bound for their numbers of real solutions which is sharp in some instances. This upper bound is nontrivial 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 Z n, this bound is 2n + 1, while the Khovanskii bound is exponential in n 2. The bound 2n + 1 is sharp and can be attained only for nondegenerate circuits. Our methods are based on computing an eliminant and involve a mixture of combinatorics, geometry, and arithmetic.
A sharper estimate on the Betti numbers of sets defined by quadratic inequalities
 Discrete and Computational Geometry, to appear. SAUGATA BASU, DMITRII V. PASECHNIK, AND MARIEFRANÇOISE
"... In this paper we consider the problem of bounding the Betti numbers, bi(S), of a semialgebraic set S ⊂ R k defined by polynomial inequalities P1 ≥ 0,..., Ps ≥ 0, where Pi ∈ R[X1,..., Xk] , s < k, and deg(Pi) ≤ 2, for 1 ≤ i ≤ s. We prove that for 0 ≤ i ≤ k − 1, bi(S) ≤ 1 1 + (k − s) + ..."
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Cited by 9 (6 self)
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In this paper we consider the problem of bounding the Betti numbers, bi(S), of a semialgebraic set S ⊂ R k defined by polynomial inequalities P1 ≥ 0,..., Ps ≥ 0, where Pi ∈ R[X1,..., Xk] , s < k, and deg(Pi) ≤ 2, for 1 ≤ i ≤ s. We prove that for 0 ≤ i ≤ k − 1, bi(S) ≤ 1 1 + (k − s) +
A quantifier elimination algorithm for linear real arithmetic
 In LPAR (Logic for Programming, Artificial Intelligence, and Reasoning), LNCS
"... We propose a new quantifier elimination algorithm for the theory of linear real arithmetic. This algorithm uses as subroutines satisfiability modulo this theory and polyhedral projection; there are good algorithms and implementations for both of these. The quantifier elimination algorithm presented ..."
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Cited by 9 (4 self)
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We propose a new quantifier elimination algorithm for the theory of linear real arithmetic. This algorithm uses as subroutines satisfiability modulo this theory and polyhedral projection; there are good algorithms and implementations for both of these. The quantifier elimination algorithm presented in the paper is compared, on examples arising from program analysis problems and on random examples, to several other implementations, all of which cannot solve some of the examples that our algorithm solves easily. 1
Automatic modular abstractions for template numerical constraints
 Logical Methods in Computer Science
, 2010
"... We propose a method for automatically generating abstract transformers for static analysis by abstract interpretation. The method focuses on linear constraints on programs operating on rational, real or floatingpoint variables and containing linear assignments and tests. Given the specification of a ..."
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Cited by 8 (2 self)
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We propose a method for automatically generating abstract transformers for static analysis by abstract interpretation. The method focuses on linear constraints on programs operating on rational, real or floatingpoint variables and containing linear assignments and tests. Given the specification of an abstract domain, and a program block, our method transformer. It is thus a form of program transformation. In addition to loopfree code, the same method also applies for obtaining least fixed points as functions of the precondition, which permits the analysis of loops and recursive functions. The motivation of our work is dataflow synchronous programming languages, used for building controlcommand embedded systems, but it also applies to imperative and functional programming. Our algorithms are based on quantifier elimination and symbolic manipulation techniques over linear arithmetic formulas. We also give less general results for nonlinear constraints and nonlinear program constructs. 1
Optimal abstraction on realvalued programs
, 2007
"... In this paper, we show that it is possible to abstract program fragments using real variables using formulas in the theory of real closed fields. This abstraction is compositional and modular. We first propose domain (in a wide class including intervals and octagons), we then show how to obtain an o ..."
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Cited by 5 (5 self)
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In this paper, we show that it is possible to abstract program fragments using real variables using formulas in the theory of real closed fields. This abstraction is compositional and modular. We first propose domain (in a wide class including intervals and octagons), we then show how to obtain an optimal abstraction of program fragments with respect to that domain. This abstraction allows computing optimal fixed points inside that abstract domain, without the need for a widening operator. 1
The smallest multistationary masspreserving chemical reaction network
"... Abstract. Biochemical models that exhibit bistability are of interest to biologists and mathematicians alike. Chemical reaction network theory can provide sufficient conditions for the existence of bistability, and on the other hand can rule out the possibility of multiple steady states. Understandi ..."
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Cited by 2 (1 self)
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Abstract. Biochemical models that exhibit bistability are of interest to biologists and mathematicians alike. Chemical reaction network theory can provide sufficient conditions for the existence of bistability, and on the other hand can rule out the possibility of multiple steady states. Understanding small networks is important because the existence of multiple steady states in a subnetwork of a biochemical model can sometimes be lifted to establish multistationarity in the larger network. This paper establishes the smallest reversible, masspreserving network that admits bistability and determines the semialgebraic set of parameters for which more than one steady state exists.
Optimizing nvariate (n+k)nomials for small k
, 2010
"... We give a high precision polynomialtime approximation scheme for the supremum of any honest nvariate (n + 2)nomial with a constant term, allowing real exponents as well as real coefficients. Our complexity bounds count field operations and inequality checks, and are quadratic in n and the logarit ..."
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Cited by 1 (1 self)
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We give a high precision polynomialtime approximation scheme for the supremum of any honest nvariate (n + 2)nomial with a constant term, allowing real exponents as well as real coefficients. Our complexity bounds count field operations and inequality checks, and are quadratic in n and the logarithm of a certain condition number. For the special case of nvariate (n+2)nomials with integer exponents, the log of our condition number is subquadratic in the sparse size. The best previous complexity bounds were exponential in the sparse size, even for n fixed. Along the way, we partially extend the theory of Viro diagrams and Adiscriminants to real exponents. We also show that, for any fixed δ>0, deciding whether the supremum of an nvariate ( n+n δ)nomial exceeds a given number is NPRcomplete.