Multivariate Interpolation and Standard Bases for Macaulay Modules

Cerlienco, Luigi; Mureddu, Marina

doi:10.1006/jabr.2001.9061

Cited by 11 publications

(16 citation statements)

References 10 publications

(12 reference statements)

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“…Such correspondence is the output of a combinatorial algorithm proposed by Cerlienco and Mureddu [14,15,16] and produces exactly the same values N(I(X σ )) as Möller Algorithm [29].…”

Section: Cerlienco-mureddu Correspondencementioning

confidence: 99%

“…While many algorithms are available which, given a finite set of points X produce the normal set N ⊂ T s.t. N = N(I(X) [35,14,15,16,29,22,11] and a Cerlienco-Mureddu Correspondence Φ : N → X [14,15,16,29,11], since we are interested in applying it to the elementary case n = 2, we make reference to the first and stronger instance proposed by Lazard [25]: denoting π :…”

Section: Lazard's Cerlienco-mureddu Correspondencementioning

confidence: 99%

“…. , ℓ iri }, the aim is to describe the combinatorial structure of both the ideal I = ∪ i q i and the algebra F[N(I)] = P \ I avoiding Buchberger Algorithm and Groebner bases of I and even Buchberger reduction modulo I in favour of combinatorial algorithms, as variations of Cerlienco-Mureddu Algorithm [8,11,14,15,16,22] and Lundqvist interpolation formula [27], dealing with F[N(I)] and {ℓ ij ,…”

Section: Introductionmentioning

confidence: 99%

“…the ideal of points of Z, produced with good complexity, via Cerlienco-Mureddu Algorithm [14,15,16], the set N := N(I(Z)) and proved that the related Gröbner basis has the shape…”

Section: Introductionmentioning

confidence: 99%

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Macaulay, Lazard and the Syndrome Variety

Ceria¹

2019

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In this paper we consider the four syndrom varieties Z × e , i.e. the set of all error locations corresponding to errors of weight w, 0 ≤ w ≤ 2, Z × ns , the set of all non spurious error locations corresponding to errors of weight w, 0 ≤ w ≤ 2, Z × + , the set of all non-spurious error locations corresponding to errors of weight w, 1 ≤ w ≤ 2, Z × 2 , the set of all non-spurious error locations corresponding to errors of weight w = 2, associated to an up-to-two errors correcting binary cyclic codes. Denoting J * := I(Z * ), the ideal of these syndrome varieties, N * := N(J * ) the Gröbner escalier of J * w.r.t. the lex ordering with x1 < x2 < z1 < z2, Φ * : Z * → N * a Cerlienco-Mureddu correspondence, and G * a minimal Groebner basis of the ideal J * , the aim of the paper is, assuming to know the structure of the order ideal N2 and a Cerlienco Mureddu Correspondence to deduce with elementary arguments N * , G * and Φ * for * ∈ {e, ns, +}.The tools are Macaulay's trick and Lazard's formulation of Cerlienco-Mureddu correspondence. * The author expresses her heartful thanks to Teo Mora for the many fruitful discussions and suggestions on this topic. 1 The recent mood, not depending on the network technology of the Sixties, and thus not needing the key equation, prefers consider the plain error locator polynomial µ j=1 (x − α ℓ j ).

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“…Such correspondence is the output of a combinatorial algorithm proposed by Cerlienco and Mureddu [14,15,16] and produces exactly the same values N(I(X σ )) as Möller Algorithm [29].…”

Section: Cerlienco-mureddu Correspondencementioning

confidence: 99%

Section: Lazard's Cerlienco-mureddu Correspondencementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…the ideal of points of Z, produced with good complexity, via Cerlienco-Mureddu Algorithm [14,15,16], the set N := N(I(Z)) and proved that the related Gröbner basis has the shape…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Macaulay, Lazard and the Syndrome Variety

Ceria¹

2019

Preprint

View full text Add to dashboard Cite

show abstract

“…A very simple basis for A is the lexicographical Groebner escalier N(X) of I and it can be computed in a purely combinatorial way, without using Groebner bases (see [7,11,12,13,14,24]). Once one has the escalier, it is a trivial task to find a generating set U for T(I).…”

Section: Janet-like Division If and Only If The Following Conditions ...mentioning

confidence: 99%