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23
A Gröbner free alternative for polynomial system solving
 Journal of Complexity
, 2001
"... Given a system of polynomial equations and inequations with coefficients in the field of rational numbers, we show how to compute a geometric resolution of the set of common roots of the system over the field of complex numbers. A geometric resolution consists of a primitive element of the algebraic ..."
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Cited by 109 (19 self)
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Given a system of polynomial equations and inequations with coefficients in the field of rational numbers, we show how to compute a geometric resolution of the set of common roots of the system over the field of complex numbers. A geometric resolution consists of a primitive element of the algebraic extension defined by the set of roots, its minimal polynomial and the parametrizations of the coordinates. Such a representation of the solutions has a long history which goes back to Leopold Kronecker and has been revisited many times in computer algebra. We introduce a new generation of probabilistic algorithms where all the computations use only univariate or bivariate polynomials. We give a new codification of the set of solutions of a positive dimensional algebraic variety relying on a new global version of Newton’s iterator. Roughly speaking the complexity of our algorithm is polynomial in some kind of degree of the system, in its height, and linear in the complexity of evaluation
LOWER BOUNDS FOR DIOPHANTINE APPROXIMATIONS
, 1996
"... We introduce a subexponential algorithm for geometric solving of multivariate polynomial equation systems whose bit complexity depends mainly on intrinsic geometric invariants of the solution set. ¿From this algorithm, we derive a new procedure for the decision of consistency of polynomial equation ..."
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Cited by 70 (27 self)
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We introduce a subexponential algorithm for geometric solving of multivariate polynomial equation systems whose bit complexity depends mainly on intrinsic geometric invariants of the solution set. ¿From this algorithm, we derive a new procedure for the decision of consistency of polynomial equation systems whose bit complexity is subexponential, too. As a byproduct, we analyze the division of a polynomial modulo a reduced complete intersection ideal and from this, we obtain an intrinsic lower bound for the logarithmic height of diophantine approximations to a given solution of a zero–dimensional polynomial equation system. This result represents a multivariate version of Liouville’s classical theorem on approximation of algebraic numbers by rationals. A special feature of our procedures is their polynomial character with respect to the mentioned geometric invariants when instead of bit operations only arithmetic operations are counted at unit cost. Technically our paper relies on the use of straight–line programs as a data structure for the encoding of polynomials, on a new symbolic application of Newton’s algorithm to the Implicit Function Theorem and on a special, basis independent trace formula for affine Gorenstein algebras.
When polynomial equation systems can be "solved" fast?
 IN PROC. 11TH INTERNATIONAL SYMPOSIUM APPLIED ALGEBRA, ALGEBRAIC ALGORITHMS AND ERRORCORRECTING CODES, AAECC11
, 1995
"... We present a new method for solving symbolically zerodimensional polynomial equation systems in the affine and toric case. The main feature of our method is the use of an alternative data structure: arithmetic networks and straightline programs with FOR gates. For sequential time complexity measu ..."
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Cited by 66 (20 self)
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We present a new method for solving symbolically zerodimensional polynomial equation systems in the affine and toric case. The main feature of our method is the use of an alternative data structure: arithmetic networks and straightline programs with FOR gates. For sequential time complexity measured by the size of these networks we obtain the following result: it is possible to solve any affine or toric zerodimensional equation system in nonuniform sequential time which is polynomial in the length of the input description and the &quot;geometric degree &quot; of the equation system. Here, the input is thought to be given by a straightline program (or alternatively in sparse representation), and the length of the input is measured by number of variables, degree of equations and size of the program (or sparsity of the equations). Geometric degree has to be adequately defined. It is always bounded by the algebraiccombinatoric &quot;B'ezout number &quot; of the system which is given by the Hilbert function of a suitable homogeneous ideal. However, in many important cases, the value of the geometric degree is much smaller than
Straightline programs in geometric elimination theory
 J. Pure Appl. Algebra
, 1998
"... Dedicated to Volker Strassen for his work on complexity We present a new method for solving symbolically zero–dimensional polynomial equation systems in the affine and toric case. The main feature of our method is the use of problem adapted data structures: arithmetic networks and straight–line prog ..."
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Cited by 65 (17 self)
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Dedicated to Volker Strassen for his work on complexity We present a new method for solving symbolically zero–dimensional polynomial equation systems in the affine and toric case. The main feature of our method is the use of problem adapted data structures: arithmetic networks and straight–line programs. For sequential time complexity measured by network size we obtain the following result: it is possible to solve any affine or toric zero–dimensional equation system in non–uniform sequential time which is polynomial in the length of the input description and the “geometric degree ” of the equation system. Here, the input is thought to be given by a straight–line program (or alternatively in sparse representation), and the length of the input is measured by number of variables, degree of equations and size of the program (or sparsity of the equations). The geometric degree of the input system has to be adequately defined. It is always bounded by the algebraic–combinatoric “Bézout number ” of the system which is given by the Hilbert function of a suitable homogeneous ideal. However, in many important cases, the value of the geometric
Sharp estimates for the arithmetic Nullstellensatz
 Duke Math. J
"... We present sharp estimates for the degree and the height of the polynomials in the Nullstellensatz over the integer ring Z. The result improves previous work of P. Philippon, C. Berenstein and A. Yger, and T. Krick and L. M. Pardo. We also present degree and height estimates of intrinsic type, which ..."
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Cited by 34 (4 self)
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We present sharp estimates for the degree and the height of the polynomials in the Nullstellensatz over the integer ring Z. The result improves previous work of P. Philippon, C. Berenstein and A. Yger, and T. Krick and L. M. Pardo. We also present degree and height estimates of intrinsic type, which depend mainly on the degree and the height of the input polynomial system. As an application we derive an effective arithmetic Nullstellensatz for sparse polynomial systems. The proof of these results relies heavily on the notion of local height of an affine
Polar Varieties and Efficient Real Equation Solving: The Hypersurface Case
 PROCEEDINGS OF THE 3RD CONFERENCE APPROXIMATION AND OPTIMIZATION IN THE CARIBBEAN, IN: APORTACIONES MATEMÁTICAS, MEXICAN SOCIETY OF MATHEMATICS
, 1998
"... The objective of this paper is to show how the recently proposed method by Giusti, Heintz, Morais, Morgenstern, Pardo [10] can be applied to a case of real polynomial equation solving. Our main result concerns the problem of finding one representative point for each connected component of a real bou ..."
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Cited by 33 (5 self)
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The objective of this paper is to show how the recently proposed method by Giusti, Heintz, Morais, Morgenstern, Pardo [10] can be applied to a case of real polynomial equation solving. Our main result concerns the problem of finding one representative point for each connected component of a real bounded smooth hypersurface. The algorithm in [10] yields a method for symbolically solving a zerodimensional polynomial equation system in the affine (and toric) case. Its main feature is the use of adapted data structure: Arithmetical networks and straightline programs. The algorithm solves any affine zerodimensional equation system in nonuniform sequential time that is polynomial in the length of the input description and an adequately defined affine degree of the equation system. Replacing the affine degree of the equation system by a suitably defined real degree of certain polar varieties associated to the input equation, which describes the hypersurface under consideration, and using straightline program codification of the input and intermediate results, we obtain a method for the problem introduced above that is polynomial in the input length and the real degree.
The Hardness of Polynomial Equation Solving
, 2003
"... Elimination theory is at the origin of algebraic geometry in the 19th century and deals with algorithmic solving of multivariate polynomial equation systems over the complex numbers, or, more generally, over an arbitrary algebraically closed field. In this paper we investigate the intrinsic seq ..."
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Cited by 26 (14 self)
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Elimination theory is at the origin of algebraic geometry in the 19th century and deals with algorithmic solving of multivariate polynomial equation systems over the complex numbers, or, more generally, over an arbitrary algebraically closed field. In this paper we investigate the intrinsic sequential time complexity of universal elimination procedures for arbitrary continuous data structures encoding input and output objects of elimination theory (i.e. polynomial equation systems) and admitting the representation of certain limit objects.
Rational invariants of a group action. Construction and rewriting
, 2007
"... Geometric constructions applied to a rational action of an algebraic group lead to a new algorithm for computing rational invariants. A finite generating set of invariants appears as the coefficients of a reduced Gröbner basis. The algorithm comes in two variants. In the first construction the ideal ..."
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Cited by 23 (8 self)
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Geometric constructions applied to a rational action of an algebraic group lead to a new algorithm for computing rational invariants. A finite generating set of invariants appears as the coefficients of a reduced Gröbner basis. The algorithm comes in two variants. In the first construction the ideal of the graph of the action is considered. In the second one the ideal of a crosssection is added to the ideal of the graph. Zerodimensionality of the resulting ideal brings a computational advantage. In both cases, reduction with respect to the computed Gröbner basis allows us to express any rational invariant in terms of the generators.
Some Complexity Results for Polynomial Ideals
, 1997
"... In this paper, we survey some of our new results on the complexity of a number of problems related to polynomial ideals. We consider multivariate polynomials over some ring, like the integers or the rationals. For instance, a polynomial ideal membership problem is a (w + 1)tuple P = ( f, g1, g2,.. ..."
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Cited by 21 (0 self)
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In this paper, we survey some of our new results on the complexity of a number of problems related to polynomial ideals. We consider multivariate polynomials over some ring, like the integers or the rationals. For instance, a polynomial ideal membership problem is a (w + 1)tuple P = ( f, g1, g2,..., gw) where f and the gi are multivariate polynomials, and the problem is to determine whether f is in the ideal generated by the gi. For polynomials over the integers or rationals, this problem is known to be exponential space complete. We discuss further complexity results for problems related to polynomial ideals, like the word and subword problems for commutative semigroups, a quantitative version of Hilbert’s Nullstellensatz in a complexity theoretic version, and problems concerning the computation of reduced polynomials and Gröbner bases.
Variations by complexity theorists on three themes of Euler, . . .
 COMPUTATIONAL COMPLEXITY
, 2005
"... This paper surveys some connections between geometry and complexity. A main role is played by some quantities —degree, Euler characteristic, Betti numbers — associated to algebraic or semialgebraic sets. This role is twofold. On the one hand, lower bounds on the deterministic time (sequential and pa ..."
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Cited by 15 (3 self)
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This paper surveys some connections between geometry and complexity. A main role is played by some quantities —degree, Euler characteristic, Betti numbers — associated to algebraic or semialgebraic sets. This role is twofold. On the one hand, lower bounds on the deterministic time (sequential and parallel) necessary to decide a set S are established as functions of these quantities associated to S. The optimality of some algorithms is obtained as a consequence. On the other hand, the computation of these quantities gives rise to problems which turn out to be hard (or complete) in different complexity classes. These two kind of results thus turn the quantities above into measures of complexity in two quite different ways.