Modular strategic SMT solving with SMT-RAT

Kremer, Gereon; Ábrahám, Erika

doi:10.2478/ausi-2018-0001

Cited by 1 publication

(2 citation statements)

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“…Our experiments have shown the potential of a good implementation, and how this opens the way to new applications. For example in Abbott et al (2018a), we use our primary decomposition approach for factoring polynomials over algebraic field extensions in advanced methods in the context of the SC 2 community: it is proving to be useful in the software CArL/SMT-RAT by Kremer and Ábrahám (2018) which implements Lazard's variant of Cylindrical Algebraic Decomposition.…”

Section: Discussionmentioning

confidence: 99%

“…For instance, one fundamental tool in the algebraic approach to tackling SC 2 problems is Cylindrical Algebraic Decomposition, often abbreviated to CAD (first introduced in (Collins, 1975)). The software CArL/SMT-RAT by Kremer and Ábrahám (2018) employs Lazard's variant of CAD (McCallum et al, 2017) which requires polynomial factorization over algebraic field extensions. We use our efficient algorithms for minimal polynomials to compute quickly the primary decomposition of zero-dimensional ideals (see Section 4.5 and Table 5), which in turn give a good method for factorizing polynomials over algebraic field extensions.…”

Section: Relevance To the Sc 2 Communitymentioning

confidence: 99%

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Computing and using minimal polynomials

Abbott

Bigatti

Palezzato

et al. 2020

Journal of Symbolic Computation

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Given a zero-dimensional ideal I in a polynomial ring, many computations start by finding univariate polynomials in I. Searching for a univariate polynomial in I is a particular case of considering the minimal polynomial of an element in P/I. It is well known that minimal polynomials may be computed via elimination, therefore this is considered to be a "resolved problem". But being the key of so many computations, it is worth investigating its meaning, its optimization, its applications (e.g. testing if a zero-dimensional ideal is radical, primary or maximal). We present efficient algorithms for computing the minimal polynomial of an element of P/I. For the specific case where the coefficients are in Q, we show how to use modular methods to obtain a guaranteed result. We also present some applications of minimal polynomials, namely algorithms for computing radicals and primary decompositions of zero-dimensional ideals, and also for testing radicality and maximality.

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Section: Discussionmentioning

confidence: 99%