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26
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 18 (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.
Probabilistic Algorithms for Geometric Elimination
 in Engineering, Communication and Computing
, 1999
"... We develop probabilistic algorithms that solve problems of geometric elimination theory using small memory resources. These algorithms are obtained by means of the adaptation of a general transformation due to A. Borodin which converts uniform boolean circuit depth into sequential (Turing machine) s ..."
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Cited by 12 (5 self)
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We develop probabilistic algorithms that solve problems of geometric elimination theory using small memory resources. These algorithms are obtained by means of the adaptation of a general transformation due to A. Borodin which converts uniform boolean circuit depth into sequential (Turing machine) space. The boolean circuits themselves are developed using techniques based on the computation of a primitive element of a suitable zerodimensional algebra and diophantine considerations. Our algorithms improve...
A New Lower Bound Construction for Commutative Thue Systems, with Applications
, 1997
"... For n 1; d 2, we describe a commutative Thue system that has ¸ 2n variables and O(n) rules, each rule of size d + O(1) and that counts to d 2 n in a certain technical sense. This gives a more "efficient" alternative to a wellknown construction of Mayr and Meyer. Using this constructi ..."
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Cited by 11 (1 self)
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For n 1; d 2, we describe a commutative Thue system that has ¸ 2n variables and O(n) rules, each rule of size d + O(1) and that counts to d 2 n in a certain technical sense. This gives a more "efficient" alternative to a wellknown construction of Mayr and Meyer. Using this construction, we sharpen the known doubleexponential lower bounds for the maximum degrees D(n; d); I(n; d); S(n; d) associated (respectively) with Grobner bases, ideal membership problem and the syzygy basis problem: D(n; d) S(n; d) d 2 m ; I(n; d) d 2 m where m ¸ n=2, and n; d sufficiently large. For comparison, it was known that D(n; d) d 2 n and I(n; d) (2d) 2 n . Keywords: Lower bound, commutative Thue system, Grobner basis, ideal membership problem, syzygy basis problem This research is supported in part by NSF Grants DCR8401898, CCR8703458 and ONR Grant N0001485K0046. The work was carried out while visiting the Research Institute for Symbolic Computation (RISCLINZ), Johannes K...
An Optimal Algorithm for Constructing the Reduced Gröbner Basis of Binomial Ideals
 J. SYMBOLIC COMPUT
, 1996
"... In this paper, we present an optimal, exponential space algorithm for generating the reduced Gröbner basis of binomial ideals. We make use of the close relationship between commutative semigroups and pure difference binomial ideals. Based on the algorithm for the uniform word problem in commutative ..."
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Cited by 9 (5 self)
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In this paper, we present an optimal, exponential space algorithm for generating the reduced Gröbner basis of binomial ideals. We make use of the close relationship between commutative semigroups and pure difference binomial ideals. Based on the algorithm for the uniform word problem in commutative semigroups exhibited by Mayr and Meyer we first derive an exponential space algorithm for constructing the reduced Gröbner basis of a pure difference binomial ideal. In addition to some applications to finitely presented commutative semigroups, this algorithm is then extended to an exponential space algorithm for generating the reduced Gröbner basis of binomial ideals in general.
Exponential space computation of Gröbner bases
, 1996
"... Given a polynomial ideal and a term order, there is a unique reduced Gröbner basis and, for each polynomial, a unique normal form, namely the smallest (w.r.t. the term order) polynomial in the same coset. We consider the problem of finding this normal form for any given polynomial, without prior ..."
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Cited by 7 (3 self)
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Given a polynomial ideal and a term order, there is a unique reduced Gröbner basis and, for each polynomial, a unique normal form, namely the smallest (w.r.t. the term order) polynomial in the same coset. We consider the problem of finding this normal form for any given polynomial, without prior computation of the Gröbner basis. This is done by transforming a representation of the normal form into a system of linear equations and solving this system. Using the ability to find normal forms, we show how to obtain the Grobner basis in exponential space.
Reduction mod p of standard bases
 Comm. Algebra
, 2005
"... We investigate the behavior of standard bases (in the sense of Hironaka and Grauert) for ideals in rings of formal power series over commutative rings with respect to specializations of the coefficients. For instance, we show that any ideal I of the ring of formal power series A[[X]] = A[[X1,..., ..."
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Cited by 6 (0 self)
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We investigate the behavior of standard bases (in the sense of Hironaka and Grauert) for ideals in rings of formal power series over commutative rings with respect to specializations of the coefficients. For instance, we show that any ideal I of the ring of formal power series A[[X]] = A[[X1,..., XN]] with coefficients in a Noetherian ring A admits a standard basis whose image under every specialization of A onto a field is a standard basis of the image of I. Applications include a modular criterion for ideal membership in Z[[X]] and a constructibility result for ideal membership in K[[X]], where K is a field.
ORDERINGS OF MONOMIAL IDEALS
, 2003
"... We study the set of monomial ideals in a polynomial ring as an ordered set, with the ordering given by reverse inclusion. We give a short proof of the fact that every antichain of monomial ideals is finite. Then we investigate ordinal invariants for the complexity of this ordered set. In particular ..."
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Cited by 4 (1 self)
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We study the set of monomial ideals in a polynomial ring as an ordered set, with the ordering given by reverse inclusion. We give a short proof of the fact that every antichain of monomial ideals is finite. Then we investigate ordinal invariants for the complexity of this ordered set. In particular, we give an interpretation of the height function in terms of the HilbertSamuel polynomial, and we compute upper and lower bounds on the maximal order type.
Groebner Basis Methods for Multichannel Sampling with Unknown Offsets
, 2008
"... In multichannel sampling, several sets of subNyquist sampled signal values are acquired. The offsets between the sets are unknown, and have to be resolved, just like the parameters of the signal itself. This problem is nonlinear in the offsets, but linear in the signal parameters. We show that when ..."
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Cited by 3 (2 self)
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In multichannel sampling, several sets of subNyquist sampled signal values are acquired. The offsets between the sets are unknown, and have to be resolved, just like the parameters of the signal itself. This problem is nonlinear in the offsets, but linear in the signal parameters. We show that when the basis functions for the signal space are related to polynomials, we can express the joint offset and signal parameter estimation as a set of polynomial equations. This is the case for example with polynomial signals or Fourier series. The unknown offsets and signal parameters can be computed exactly from such a set of polynomials using Gröbner bases and Buchberger’s algorithm. This solution method is developed in detail after a short and tutorial overview of Gröbner basis methods. We then address the case of noisy samples, and consider the computational complexity, exploring simplifications due to the special structure of the problem.
The Geometry in Constraint Logic Programs
 In Position Papers for the First Workshop on Principles and Practice of Constraint Programming
, 1993
"... Many applications of constraint programming languages concern geometric domains. We propose incorporating strong algorithmic techniques from the study of geometric and algebraic algorithms into the implementation of constraint programming languages. Interesting new computational problems in computat ..."
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Cited by 3 (1 self)
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Many applications of constraint programming languages concern geometric domains. We propose incorporating strong algorithmic techniques from the study of geometric and algebraic algorithms into the implementation of constraint programming languages. Interesting new computational problems in computational geometry and computer algebra arises from such considerations. We look at what is known and what needs to be addressed.