Decomposition of polynomial sets into characteristic pairs

Wang, Dongming; Dong, Rina; Mou, Chenqi

doi:10.48550/arxiv.1702.08664

Cited by 1 publication

(5 citation statements)

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“…, (C e , G e ) such that The triangular sets in an src decomposition are normal in most cases (cf. [32,31]). It turns out that comprehensive triangular decompositions [8] and/or Gröbner systems [23] can be reproduced rather easily from src pairs computed by our algorithm, and we are working on the details.…”

Section: Discussionmentioning

confidence: 99%

“…Some structural properties about pseudo-divisibility among polynomials in the Gröbner bases can be found in [31]. Based on those properties, an effective algorithm for normal triangular decomposition of polynomial sets has been proposed in [32].…”

Section: W-characteristic Sets Characteristic Pairs and Characteristi...mentioning

confidence: 99%

“…When the regular or normal characteristic pairs (C i , G i ) are computed by the algorithms described in [29,32], the zero relation…”

Section: W-characteristic Sets Characteristic Pairs and Characteristi...mentioning

confidence: 99%

“…It is widely known that the theories and methods of triangular sets are different from those of Gröbner bases conceptually and operationally. The questions that have motivated our work here and in [31,32] are what inherent relationship there may exist between triangular sets and Gröbner bases and how to connect or combine the two algorithmic approaches to amplify their applicability and power. These questions have been touched primarily for some special polynomial ideals such as bivariate ideals [18] and zero-dimensional ideals [22,12,19], while the general literature of studies on triangular sets and Gröbner bases is extremely rich (see [2,4,6,7,8,9,12,13,14,15,17,18,20,21,23,24,25,26,28,29,31,32] and references therein).…”

Section: Introductionmentioning

confidence: 99%

“…Nevertheless, for each sat(T i ) a Gröbner basis can be computed straightforwardly from T i . Another alternative approach proposed recently by Mou and the authors [32] permits one to decompose an arbitrary polynomial set into normal or regular characteristic pairs directly. This approach is based on a structure theorem about irregular W-characteristic sets [31] for the splitting of ideals and relies strongly upon Gröbner basis computation; it is independent of pseudo-division-based triangular decomposition.…”

Section: Introductionmentioning

confidence: 99%

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Computing strong regular characteristic pairs with Groebner bases

Dong,

Wang

2019

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The W-characteristic set of a polynomial ideal is the minimal triangular set contained in the reduced lexicographical Gröbner basis of the ideal. A pair (C, G) of polynomial sets is a strong regular characteristic pair if G is a reduced lexicographical Gröbner basis, C is the W-characteristic set of the ideal G , the saturated ideal sat(C) of C is equal to G , and C is regular. In this paper, we show that for any regular set T the W-characteristic set C and the reduced lexicographical Gröbner basis G of sat(T ) form a strong regular characteristic pair (C, G), and for any polynomial ideal I with given generators one can either detect that I is unit, or construct a strong regular characteristic pair (C, G) by computing Gröbner bases such that I ⊆ sat(C), so the radical ideal of I can be split into the saturated ideal sat(C) and the quotient ideal I : sat(C). Based on this strategy of splitting by means of quotient and with Gröbner basis and ideal computations, we devise a simple algorithm to decompose an arbitrary polynomial set F into finitely many strong regular characteristic pairs, from which two representations for the zeros of F are obtained: one in terms of strong regular Gröbner bases and the other in terms of regular triangular sets. We present some properties about strong regular characteristic pairs and characteristic decompositions and illustrate the proposed algorithm and its performance by examples and experimental results.

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Section: Discussionmentioning

confidence: 99%

Section: W-characteristic Sets Characteristic Pairs and Characteristi...mentioning

confidence: 99%

“…When the regular or normal characteristic pairs (C i , G i ) are computed by the algorithms described in [29,32], the zero relation…”

Section: W-characteristic Sets Characteristic Pairs and Characteristi...mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Computing strong regular characteristic pairs with Groebner bases

Dong,

Wang

2019

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

Decomposition of polynomial sets into characteristic pairs

Cited by 1 publication

References 26 publications

Computing strong regular characteristic pairs with Groebner bases

Computing strong regular characteristic pairs with Groebner bases

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