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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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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.
Ideal Membership in Polynomial Rings over the Integers
 J. Amer. Math. Soc
"... Abstract. We present a new approach to the ideal membership problem for polynomial rings over the integers: given polynomials f0, f1,..., fn ∈ Z[X], where X = (X1,..., XN) is an Ntuple of indeterminates, are there g1,..., gn ∈ Z[X] such that f0 = g1f1 + · · · + gnfn? We show that the degree of th ..."
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Abstract. We present a new approach to the ideal membership problem for polynomial rings over the integers: given polynomials f0, f1,..., fn ∈ Z[X], where X = (X1,..., XN) is an Ntuple of indeterminates, are there g1,..., gn ∈ Z[X] such that f0 = g1f1 + · · · + gnfn? We show that the degree of the polynomials g1,..., gn can be bounded by (2d) 2O(N2) (h + 1) where d is the maximum total degree and h the maximum height of the coefficients of f0,..., fn. Some related questions, primarily concerning linear equations in R[X], where R is the ring of integers of a number field, are also treated.
DBases for Polynomial Ideals over Commutative Noetherian Rings
 8TH INTL. CONF. ON REWRITING TECHNIQUES AND APPLICATIONS
, 1997
"... We present a completionlike procedure for constructing Dbases for polynomial ideals over commutative Noetherian rings with unit. The procedure is described at an abstract level, by transition rules. Its termination is proved under certain assumptions about the strategy that controls the applica ..."
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Cited by 5 (3 self)
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We present a completionlike procedure for constructing Dbases for polynomial ideals over commutative Noetherian rings with unit. The procedure is described at an abstract level, by transition rules. Its termination is proved under certain assumptions about the strategy that controls the application of the transition rules. Correctness is established by proof simplication techniques.
Integrated Development of Algebra in Type Theory
, 1998
"... We present the project of developing computational algebra inside type theory in an integrated way. As a first step towards this, we present direct constructive proofs of Dickson's lemma and Hilbert's basis theorem, and use this to prove the constructive existence of Grobner bases. This can be se ..."
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We present the project of developing computational algebra inside type theory in an integrated way. As a first step towards this, we present direct constructive proofs of Dickson's lemma and Hilbert's basis theorem, and use this to prove the constructive existence of Grobner bases. This can be seen as an integrated development of the Buchberger algorithm, and so far we have a concise formalisation of Dickson's lemma in Half, a type checker for a variant of MartinLof's type theory. We then present work in progress on understanding commutative algebra constructively in type theory using formal topology. Currently we are interested in interpreting existence proofs of prime and maximal ideals, and valuation rings. We give two casestudies: a proof that certain a are nilpotent which uses prime ideals, and a proof of Dedekind's Prague theorem which uses valuation rings. 1 Introduction For the development and formal verification of algorithms, there are essentially two methods [...
IDEALS IN COMPUTABLE RINGS
"... Abstract. We show that the existence of a nontrivial proper ideal in a commutative ring with identity which is not a field is equivalent to WKL0 over RCA0, and that the existence of a nontrivial proper finitely generated ideal in a commutative ring with identity which is not a field is equivalent to ..."
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Abstract. We show that the existence of a nontrivial proper ideal in a commutative ring with identity which is not a field is equivalent to WKL0 over RCA0, and that the existence of a nontrivial proper finitely generated ideal in a commutative ring with identity which is not a field is equivalent to ACA0 over RCA0. We also prove that there are computable commutative rings with identity where the nilradical is Σ 0 1complete, and the Jacobson radical is Π 0 2complete, respectively. 1.
Finitization Procedures and Finite Model Property.
"... Investigations into the Relevant and Paraconsistent model theory of rstorder arithmetic have provided interesting new methods and results which have revived the interest in Hilbert's program. The attempt to develop Strict Finitist Mathematics using G. Priest's Collapsing lemma to nitize innite m ..."
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Investigations into the Relevant and Paraconsistent model theory of rstorder arithmetic have provided interesting new methods and results which have revived the interest in Hilbert's program. The attempt to develop Strict Finitist Mathematics using G. Priest's Collapsing lemma to nitize innite models is an example. In the investigation of some systems of Relevant Logics, another nitization procedure is used to solve positively their decision problem and to prove the nite model property for these systems. Some results related to the procedure used in these investigations show that Hilbert's ideal cannot be entirely fullled or that it must be reinterpreted. 1
On Polynomial Ideals, Their Complexity, and Applications
, 1995
"... A polynomial ideal membership problem is a (w+1)tuple P = (f; g 1 ; g 2 ; : : : ; g w ) where f and the g i are multivariate polynomials over some ring, and the problem is to determine whether f is in the ideal generated by the g i . For polynomials over the integers or rationals, it is known th ..."
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A polynomial ideal membership problem is a (w+1)tuple P = (f; g 1 ; g 2 ; : : : ; g w ) where f and the g i are multivariate polynomials over some ring, and the problem is to determine whether f is in the ideal generated by the g i . For polynomials over the integers or rationals, it is known that this problem is exponential space complete.