Results 1  10
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18
Algebraic Algorithms for Sampling from Conditional Distributions
 Annals of Statistics
, 1995
"... We construct Markov chain algorithms for sampling from discrete exponential families conditional on a sufficient statistic. Examples include generating tables with fixed row and column sums and higher dimensional analogs. The algorithms involve finding bases for associated polynomial ideals and so a ..."
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Cited by 271 (20 self)
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We construct Markov chain algorithms for sampling from discrete exponential families conditional on a sufficient statistic. Examples include generating tables with fixed row and column sums and higher dimensional analogs. The algorithms involve finding bases for associated polynomial ideals and so an excursion into computational algebraic geometry.
Gröbner Bases for Binomials with Parametric Exponents
 TECHNISCHE UNIVERSITÄT MÜNCHEN
, 2004
"... We study the uniformity of Buchberger algorithms for computing Gröbner bases with respect to a natural number parameter k in the exponents of the input polynomials. The problem is motivated by positive results of T. Takahashi on special exponential parameter series of polynomial sets in singular ..."
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Cited by 9 (0 self)
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We study the uniformity of Buchberger algorithms for computing Gröbner bases with respect to a natural number parameter k in the exponents of the input polynomials. The problem is motivated by positive results of T. Takahashi on special exponential parameter series of polynomial sets in singularity theory. For arbitrary input sets uniformity is in general impossible. By way of contrast we show that the Buchberger algorithm is indeed uniform up to a finite case distinction on the exponential parameter k for inputs consisting of monomials and binomials only. Under this hypothesis the case distinction is algorithmic and partitions the parameter range into Presburger sets. In each case the Buchberger algorithm is uniform and can be described explicitly and algorithmically. In the course of the algorithm the exponential parameter k enters also the coefficients as exponent. Thus the uniformity in k is established with respect to parametric exponents in both terms and coefficients. These results are obtained as a consequence of a much more general theorem concerning Buchberger algorithms for sets of monomials and binomials with arbitrary parametric coefficients and exponents, generalizing the construction of Gröbner systems.
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.
Gröbner Bases, Invariant Theory and Equivariant Dynamics
, 1996
"... This paper is about algorithmic invariant theory as it is required within equivariant ..."
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Cited by 7 (0 self)
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This paper is about algorithmic invariant theory as it is required within equivariant
Groebner Basis Under Composition II
 In Proceedings of ISSAC 96 (International Symposium on Symbolic and Algebraic Computation
, 1996
"... Composition is an operation of replacing variables in a polynomial with other polynomials. The main question of this paper is: When does composition commute with Groebner basis computation (possibly under different term orderings)? We prove that this happens if the leading terms of the composition p ..."
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Cited by 5 (1 self)
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Composition is an operation of replacing variables in a polynomial with other polynomials. The main question of this paper is: When does composition commute with Groebner basis computation (possibly under different term orderings)? We prove that this happens if the leading terms of the composition polynomials form "permuted powering". This is a sequel to another paper where we dealt with a more restricted question (that required same term ordering).
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.
On the Relation Between Gröbner and Pommaret Bases
 AAECC
, 1998
"... Abstract. The reduced Pommaret basis and the reduced Gröbner basis of an ideal I with respect to a fixed admissible term order ≺ differ in general. A necessary and sufficient criterion for the coincidence of these bases is given. ..."
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Abstract. The reduced Pommaret basis and the reduced Gröbner basis of an ideal I with respect to a fixed admissible term order ≺ differ in general. A necessary and sufficient criterion for the coincidence of these bases is given.
Optimal Gröbner Base Algorithms for Binomial Ideals
, 1996
"... Little is known about upper complexity bounds for the normal form algorithms which transform a given polynomial ideal basis into a Gröbner basis. In this paper ..."
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Cited by 2 (2 self)
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Little is known about upper complexity bounds for the normal form algorithms which transform a given polynomial ideal basis into a Gröbner basis. In this paper
The Optimal Lower Bound for Generators of Invariant Rings without Finite SAGBI Bases with Respect to Any Admissible Order
 Discrete Math. Theor. Comput. Sci
, 1999
"... mportant special cases. Recently, the nonfiniteness of SAGBI bases for *,+ . )/  1032546 /7089 with respect to any admissible order was proven in [3]. In addition, it was shown that with respect to the number of variables, *,+ : )/  )032546 /089 is the "smallest" unique exampl ..."
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mportant special cases. Recently, the nonfiniteness of SAGBI bases for *,+ . )/  1032546 /7089 with respect to any admissible order was proven in [3]. In addition, it was shown that with respect to the number of variables, *,+ : )/  )032546 /089 is the "smallest" unique example for such a ring of polynomial invariants of a permutation group. In this note, we show the existence of an invariant ring generated only by polynomial invariants with a total degree of at most ; , which has no finite SAGBI basis with respect to any admissible order. Hence, ; is the optimal lower bound, because any invariant ring generated by polynomial invariants with a total degree of at most has for trivial reasons a finite SAGBI basis. In addition, we can show that our example has with respect to this property the minimal number of variables = , if we restrict ourself to polynomi