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A class of polynomial planar vector fields with polynomial first integral
Elsevier (2015-10)We give an algorithm for deciding whether a planar polynomial differential system has a first integral which factorizes as a product of defining polynomials of curves with only one place at infinity. In the affirmative ... -
Algebraic integrability of planar polynomial vector fields by extension to Hirzebruch surfaces
Springer Nature (2022-09-18)We study algebraic integrability of complex planar polynomial vector fields X=A(x,y)(∂/∂x)+B(x,y)(∂/∂y) through extensions to Hirzebruch surfaces. Using these extensions, each vector field X determines two infinite families ...
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Discrete Equivalence of Non-positive at Infinity Plane Valuations
Springer (2021-06-26)Non-positive at infinity valuations are a class of real plane valuations which have a nice geometrical behavior. They are divided in three types. We study the dual graphs of non-positive at infinity valuations and give ...
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Evaluation codes defined by finite families of plane valuations at infinity
Springer US (2012-08)We construct evaluation codes given by weight functions defined over polynomial rings in m ≥ 2 indeterminates. These weight functions are determined by sets of m−1 weight functions over polynomial rings in two indeterminates ...
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Finite families of plane valuations: value semigroup, graded algebra and Poincaré series
(2012)In this paper, the authors are interested in some applications of valuation theory to algebraic geometry and, particularly, to singularity theory. The aim of this paper is to provide a concise survey of some aspects of the ...
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Foliations with isolated singularities on Hirzebruch surfaces
De Gruyter (2021)We study foliations ℱ on Hirzebruch surfaces Sδ and prove that, similarly to those on the projective plane, any ℱ can be represented by a bi-homogeneous polynomial affine 1-form. In case ℱ has isolated singularities, we ...
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Minimal plane valuations
American Mathematical Society (AMS) (2018-07)We consider the value u(v)=lim m->m-1 a(mL) , where a(mL) is the last value of the vanishing sequence of Hº(mL) along a divisorial or irrational valuation v centered at Op2,p ,L (respectively, P ) being a line ...
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Newton–Okounkov bodies of exceptional curve valuations
European Mathematical Society (2020-03-20)We prove that the Newton–Okounkov body associated to the flag E∙:={X=Xr⊃Er⊃{q}}, defined by the surface X and the exceptional divisor Er given by any divisorial valuation of the complex projective plane P2, with respect ...
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Non-positive and negative at infinity divisorial valuations of Hirzebruch surfaces
Springer Verlag (2020-05)We consider rational surfaces Z defined by divisorial valuations ν of Hirzebruch surfaces. We introduce concepts of non-positivity and negativity at infinity for these valuations and prove that these concepts admit nice ...
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On the characterization of algebraically integrable plane foliations
American Mathematical Society (2010)We give a characterization theorem for non-degenerate plane foliations of degree different from 1 having a rational first integral. Moreover, we prove that the degree of a non-degenerate foliation as above provides the ...
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On the classification of exceptional planar functions over Fp
Springer Netherlands (2014-12)We will present many strong partial results towards a classification of exceptional planar/PN monomial functions on finite fields. The techniques we use are the Weil bound, Bézout’s theorem, and Bertini’s theorem. -
Quantum codes from a new construction of self-orthogonal algebraic geometry codes
Springer (2020)We present new quantum codes with good parameters which are constructed from self-orthogonal algebraic geometry codes. Our method permits a wide class of curves to be used in the formation of these codes. These results ...
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Seshadri-type constants and Newton-Okounkov bodies for non-positive at infinity valuations of Hirzebruch surfaces
Taylor and Francis (2023-02-01)We consider flags E• = {X ⊃ E ⊃ {q}}, where E is an exceptional divisor defining a non-positive at infinity divisorial valuation νE of a Hirzebruch surface Fδ , q a point in E and X the surface given by νE , and determine ...
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The Abhyankar-Moh Theorem for plane valuations at infinity
Elsevier (2013)We introduce the class of plane valuations at infinity and prove an analogue to the Abhyankar–Moh (semigroup) Theorem for it.
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The cone of curves and the Cox ring of rational surfaces given by divisorial valuations
Elsevier (2016-02)We consider surfaces X defined by plane divisorial valuations ν of the quo- tient field of the local ring R at a closed point p of the projective plane P 2 over an arbitrary algebraically closed field k and ...
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The log-canonical threshold of a plane curve
Cambridge University Press (2016-05)We give an explicit formula for the log-canonical threshold of a reduced germ of plane curve. The formula depends only on the first two maximal contact values of the branches and their intersection multiplicities. We ...
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The Poincaré series of multiplier ideals of a simple complete ideal in a local ring of a smooth surface
Elsevier (2010)For a simple complete ideal p of a local ring at a closed point on a smooth complex algebraic surface, we introduce an algebraic object, named Poincaré series P<sub>p</sub>, that gathers in a unified way the jumping numbers ...
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δ-Sequences and Evaluation Codes de ned by Plane Valuations at Infinity
Oxford University Press (2008)We introduce the concept of δ-sequence. A δ-sequence ∆ generates a well-ordered semigroup S in Z2 or R. We show how to construct (and compute parameters) for the dual code of any evaluation code associated with a weight ...
