An algorithm on quasi-ordinary polynomials

García, María Emilia Alonso; Luengo-Velasco, I.; Raimondo, Mario Di

doi:10.1007/3-540-51083-4_48

Cited by 7 publications

(5 citation statements)

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“…La construction des racines est donnée par un algorithme qui généralise celui du théorème de Newton-Puiseux (théorème 2), et qui pourrait se développerà l'aide d'un logiciel de calcul formel. On peut comparer l'algorithme obtenu avec celui de [2], developpé pour le cas quasi-ordinaire.…”

Section: Introductionunclassified

“…C'est-à-dire le cône (2, 1) + pos{(1, 5), (2, 1)}. (Pour w ∈ σ 2 , on définit F (2) 2 = F(Y + U 2 V ), on vérifie que F (2) 3 = F (2) 2 (Y + V 2 ) = F(Y + V 2 + U 2 V ) = F (1) 3 , et que les exposants de la série construit φ − U 2 V sont dans le cône affine (0, 2) + pos{(2, 5), (7, 2)}. )…”

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“…2 . Le discriminant du polynôme F par rapportà Y est∆ Y F = 108U 25 V 15 + 3125U 24 V 12 + 108U 23 V 16 + 12500U 22 V 13 − 2250U 22 V 12 + 18750U 20 V 14 − 4500U 20 V 13 − 27U 20 V 12 + 12500U 18 V 15 − 2250U 18 V 14 + 1600U 17 V 9 + 3125U 16 V 16 + 1600U 15 V 10 − 256U 12 V6 …”

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Singularités quasi-ordinaires toriques et polyèdre de Newton du discriminant

Pérez

2000

Can. j. math.

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Section: Introductionunclassified

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Singularités quasi-ordinaires toriques et polyèdre de Newton du discriminant

Pérez

2000

Can. j. math.

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“…(1908), Walker (1935), Zariski (1939), Abhyankar (1969), Hironaka (1984), Lipman (1978, Villamayor (1989) and Bierstone and Milman (1991). For quasi-ordinary singularities, one might think of computing adjoints using two-dimensional Puiseaux-expansions (Alonso et al, 1988).…”

Section: Inputmentioning

confidence: 99%

Rational Parametrization of Surfaces

Schicho

1998

Journal of Symbolic Computation

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“…This analogue, provided by the Abhyankar-Jung theorem, is existential in nature; however, a constructive algorithm has been developed for the special case of n ¼ 3, see Alonso et al (1989). In Sec.…”

Section: Introductionmentioning

confidence: 99%

Root-Based Compositions of Multivariate Polynomials: Structure, Geometric Interpretations, and Decomposition Results

Mills¹,

Neuerburg²

2004

Communications in Algebra

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The concept of a composed product for univariate polynomials has been explored oextensively by Brawley, Brown, Carlitz, Gao, Mills et al. Starting with these fundamental ideas and using fractional power series representation of bivariate polynomials, the current authors Mills and Neuerburg [Mills, D., Neuerburg, K. M. A bivariate analogue to the composed product of polynomials. Algebra Colloquium (to appear)] generalize the univariate results by defining and investigating a bivariate composed sum, composed multiplication, and composed product (based on function composition) for certain classes of bivariate polynomials. In this, the sequel, we extend the generalizations to certain classes of multivariate polynomials, written as f(x 1 , . . . , x n ), over an algebraically closed field of characteristic zero. In particular, we consider a composed sum and composed multiplication defined on the class of quasiordinary polynomials. We then consider what algebraic structure is imposed by these operations. Next, we consider a generalization of the geometry associated with the class of n-quasiordinary polynomials. # Communicated by S. Wiegand. ORDER REPRINTSFinally, we utilize this geometry to show that an algorithm given by Gao and Lauder [Gao, S., Lauder, A. G. B. (2001). Decomposition of polytopes and polynomials. Discrete Comput. Geom. 26(1):89-104] to determine whether a given bivariate polynomial is absolutely irreducible over a given field can be used to ascertain whether a given trivariate polynomial that factors completely in one variable decomposes according to the composed multiplication operation.

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An algorithm on quasi-ordinary polynomials

Cited by 7 publications

References 5 publications

Singularités quasi-ordinaires toriques et polyèdre de Newton du discriminant

Singularités quasi-ordinaires toriques et polyèdre de Newton du discriminant

Rational Parametrization of Surfaces

Root-Based Compositions of Multivariate Polynomials: Structure, Geometric Interpretations, and Decomposition Results

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