The Newton Polygon of a Rational Plane Curve

DʼAndrea, Carlos; Sombra, Martín

doi:10.1007/s11786-010-0045-2

Cited by 9 publications

(4 citation statements)

References 22 publications

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“…We can suppose that not all varieties X are birational equivalent (generalization of rational concept) to A d (R). The theory of Gröebner basis [8] or the Newton Polygon [9] could be applied to tackle this problem, but, with a lot of generators, computational time blows up. The problem is hardly structured.…”

Section: Theorem 1 If the Runge-kutta Methods Is Of Order P And If F mentioning

confidence: 99%

Global Optimization for Algebraic Geometry – Computing Runge–Kutta Methods

Martino

Nicosia

2012

Lecture Notes in Computer Science

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Abstract. This research work presents a new evolutionary optimization algorithm, EVO-RUNGE-KUTTA in theoretical mathematics with applications in scientific computing. We illustrate the application of EVO-RUNGE-KUTTA, a two-phase optimization algorithm, to a problem of pure algebra, the study of the parameterization of an algebraic variety, an open problem in algebra. Results show the design and optimization of particular algebraic varieties, the RungeKutta methods of order q. The mapping between algebraic geometry and evolutionary optimization is direct, and we expect that many open problems in pure algebra will be modelled as constrained global optimization problems.

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Section: Theorem 1 If the Runge-kutta Methods Is Of Order P And If F mentioning

confidence: 99%

Global Optimization for Algebraic Geometry – Computing Runge–Kutta Methods

Martino

Nicosia

2012

Lecture Notes in Computer Science

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“…We close this introduction by pointing out a recent application of theorem 1.2, to the determination of the Newton polygon of the equation of a rational plane curve in terms of a given parameterization [DS07].…”

Section: Definition 11 ([Ps03]mentioning

confidence: 97%

A refinement of the Bernštein–Kušnirenko estimate

Philippon

Sombra

2008

Advances in Mathematics

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A theorem of Kušnirenko and Bernštein shows that the number of isolated roots of a system of polynomials in a torus is bounded above by the mixed volume of the Newton polytopes of the given polynomials, and this upper bound is generically exact. We improve on this result by introducing refined combinatorial invariants of polynomials and a generalization of the mixed volume of convex bodies: the mixed integral of concave functions. The proof is based on new techniques and results from relative toric geometry.

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“…is integrally closed so that the curve is smooth in A 1 × P 1 . The formulas for deg y f , a(f ) and b(f ) follow for instance from [6], where the authors compute the Newton polytope of a parametrised curve.…”

Section: 3mentioning

confidence: 99%

Plane curves with minimal discriminant

Simon¹,

Weimann²

2018

J. Commut. Algebra

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We give lower bounds for the degree of the discriminant with respect to y of separable polynomials f ∈ K[x, y] over an algebraically closed field of characteristic zero. Depending on the invariants involved in the lower bound, we give a geometrical characterisation of those polynomials having minimal discriminant, and give an explicit construction of all such polynomials in many cases. In particular, we show that irreducible monic polynomials with minimal discriminant coincide with coordinate polynomials. We obtain analogous partial results for the case of nonmonic or reducible polynomials by studying their GL2(K[x])-orbit and by establishing some combinatorial constraints on their Newton polytope. Our results suggest some natural extensions of the embedding line theorem of Abhyankar-Moh and of the Nagata-Coolidge problem to the case of unicuspidal curves of P 1 × P 1 .

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The Newton Polygon of a Rational Plane Curve

Cited by 9 publications

References 22 publications

Global Optimization for Algebraic Geometry – Computing Runge–Kutta Methods

Global Optimization for Algebraic Geometry – Computing Runge–Kutta Methods

A refinement of the Bernštein–Kušnirenko estimate

Plane curves with minimal discriminant

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