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36
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.
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...
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 11 (4 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...
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.
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 9 (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.
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 9 (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.
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 5 (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.
A fast algorithm for Gröbner basis conversion and its applications
 J. Symbolic Comp
, 2000
"... The Gröbner walk method converts a Gröbner basis by partitioning the computation of the basis into several smaller computations following a path in the Gröbner fan of the ideal generated by the system of equations. The method works with ideals of zerodimension as well as positive dimension. Typicall ..."
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Cited by 4 (0 self)
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The Gröbner walk method converts a Gröbner basis by partitioning the computation of the basis into several smaller computations following a path in the Gröbner fan of the ideal generated by the system of equations. The method works with ideals of zerodimension as well as positive dimension. Typically, the target point of the walking path lies on the intersection of very many cones, which ends up with initial forms of a considerable number of terms. Therefore, it is crucial to the performance of the conversion to change the target point since we have to compute a Gröbner basis with respect to the elimination term order of such large initial forms. In contrast to heuristic methods found in the literature, in this paper the author presents a deterministic method to vary the target point in order to ensure the generality of the position, i.e. we always have just a few terms in the initial forms. It turns out that this theoretical result brings a dramatic speedup in practice. We have implemented the Gröbner walk method together with the deterministic method for varying the target point in the kernel of Mathematica. Our experiments show the superlative performance of our improved Gröbner walk method in comparison with other known methods. Our best performance is 3 × 10 4 times faster than the direct computation of the reduced Gröbner basis with respect to pure lexicographic term order (using the Buchberger algorithm and the sugar cube strategy). We also discuss the complexity of the conversion algorithm and prove a degree bound for polynomials in the target Gröbner basis. In the second part of the paper, we present some applications of the conversion method for implicitization and geometric reasoning. We compare the efficiency of the improved Gröbner walk method with other methods for elimination such as multivariate resultant methods. c ○ 2000 Academic Press 1.
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.