Size of coefficients of lexicographical Groöbner bases

Dahan, Xavier

doi:10.1145/1576702.1576721

Cited by 8 publications

(7 citation statements)

References 10 publications

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“…Decomposition along the y-variables. When a lexGb generates a radical ideal, well-known Euclideanbased algorithms decompose it into triangular sets [20,9,23]. To do the same for non-radical lexGbs, two complications occur: first, Hensel lifting comes into play to compensate the division required to make nilpotent polynomials non-nilpotent, second, the output may be a lexGb not necessarily a triangular set.…”

Section: Discussionmentioning

confidence: 99%

Lexicographic Groebner bases of bivariate polynomials modulo a univariate one

Dahan

2020

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Let T (x) ∈ k[x] be a monic non-constant polynomial and write R = k[x]/ T the quotient ring. Consider two bivariate polynomials a(x, y), b(x, y) ∈ R [y]. In a first part, T = p e is assumed to be the power of an irreducible polynomial p. A new algorithm that computes a minimal lexicographic Gröbner basis of the ideal a, b, p e , is introduced. A second part extends this algorithm when T is general through the "local/global" principle realized by a generalization of "dynamic evaluation", restricted so far to a polynomial T that is squarefree. The algorithm produces splittings according to the case distinction "invertible/nilpotent", extending the usual "invertible/zero" in classic dynamic evaluation. This algorithm belongs to the Euclidean family, the core being a subresultant sequence of a and b modulo T . In particular no factorization or Gröbner basis computations are necessary. The theoretical background relies on the Lazard's structural theorem for lexicographic Gröbner bases in two variables. An implementation is realized in Magma, and extensive benchmarks are provided that show clearly the benefits, sometimes huge, of this approach compared to the computation of Gröbner bases.

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Section: Discussionmentioning

confidence: 99%

Lexicographic Groebner bases of bivariate polynomials modulo a univariate one

Dahan

2020

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“…These are well-suited to generalize the use of fast interpolation algorithms, and to derive a good running-time for the reconstruction algorithm. These explicit formulas are also necessary to obtain complexity estimates on the size of coefficients by using height theory as done in [8,6]. 3. a main new ingredient of the present article is the recursive point of view of the proof of Theorem 3.…”

Section: Theorem (Interpolation) Let V Denotes the Set Of Common Zero...mentioning

confidence: 99%

“…The proof in [6] builds explicitly a Gröbner basis from W to prove these assertions. Let us carry through the case n = 3.…”

Section: Min(i) Will Denote the Minimal Basis Of The Monomial Ideal L...mentioning

confidence: 99%

“…The structure is also useful to express the polynomials in such Gröbner bases with interpolation formulas, in function of the solution points. Such formulas allow to get reasonably sharp upper bounds on the size of coefficients, as it was achieved for some special cases of lexicographic Gröbner bases in [7,8,6]. Another application is the possibility to decompose such "non-generic" Gröbner bases into smaller ones.…”

Section: Introductionmentioning

confidence: 99%

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On lexicographic Groebner bases of radical ideals in dimension zero: interpolation and structure

Dahan¹

2012

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Due to the elimination property held by the lexicographic monomial order, the corresponding Gröbner bases display strong structural properties from which meaningful informations can easily be extracted. We study these properties for radical ideals of (co)dimension zero. The proof presented relies on a combinatorial decomposition of the finite set of points whereby iterated Lagrange interpolation formulas permit to reconstruct a minimal Gröbner basis. This is the first fully explicit interpolation formula for polynomials forming a lexicographic Gröbner basis, from which the structure property can easily be read off. The inductive nature of the proof also yield a triangular decomposition algorithm from the Gröbner basis.

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“…Related work. For the bit-size bounds, a selection of related work concerned with the RUR, triangular sets, and lexicographic Gröbner bases is [8,4,3,9]. The bounds presented are the first ones dealing with non-radical systems having a general type of singularities.…”

Section: Introductionmentioning

confidence: 99%

On the Bit-Size of Non-radical Triangular Sets

Dahan

2017

Mathematical Aspects of Computer and Information Sciences

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We present upper bounds on the bit-size of coefficients of non-radical lexicographical Gröbner bases in purely triangular form (triangular sets) of dimension zero. This extends a previous work [4], constrained to radical triangular sets; it follows the same technical steps, based on interpolation. However, key notion of height of varieties is not available for points with multiplicities; therefore the bounds obtained are less universal and depend on input data. We also introduce a related family of non-monic polynomials that have smaller coefficients, and smaller bounds. It is not obvious to compute them from the initial triangular set though.

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Size of coefficients of lexicographical Groöbner bases

Cited by 8 publications

References 10 publications

Lexicographic Groebner bases of bivariate polynomials modulo a univariate one

Lexicographic Groebner bases of bivariate polynomials modulo a univariate one

On lexicographic Groebner bases of radical ideals in dimension zero: interpolation and structure

On the Bit-Size of Non-radical Triangular Sets

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