Good subsemigroups of ℕn

2018 ◽  
Vol 28 (02) ◽  
pp. 179-206 ◽  
Author(s):  
Marco D’Anna ◽  
Pedro A. García-Sánchez ◽  
Vincenzo Micale ◽  
Laura Tozzo
Keyword(s):  
Algebraic Curve ◽  
Algebraic Curves ◽  
Good System ◽  
Canonical Ideal ◽  
Compatibility Property ◽  
Algebraic Counterpart

Value semigroups of non-irreducible singular algebraic curves and their fractional ideals are submonoids of [Formula: see text] that are closed under infimums, have a conductor and fulfill a special compatibility property on their elements. Monoids of [Formula: see text] fulfilling these three conditions are known in the literature as good semigroups and there are examples of good semigroups that are not realizable as the value semigroup of an algebraic curve. In this paper, we consider good semigroups independently from their algebraic counterpart, in a purely combinatorial setting. We define the concept of good system of generators, and we show that minimal good systems of generators are unique. Moreover, we give a constructive way to compute the canonical ideal and the Arf closure of a good subsemigroup when [Formula: see text].

scholarly journals Least Squares Fitting of Piecewise Algebraic Curves

2007 ◽  
Vol 2007 ◽  
pp. 1-11 ◽  
Author(s):  
Chun-Gang Zhu ◽  
Ren-Hong Wang
Keyword(s):  
Least Squares ◽  
Algebraic Curve ◽  
Algebraic Curves ◽  
Scattered Data ◽  
Energy Term ◽  
Bivariate Spline ◽  
Piecewise Algebraic Curve ◽  
Piecewise Algebraic Curves ◽  
The Given

A piecewise algebraic curve is defined as the zero contour of a bivariate spline. In this paper, we present a new method for fittingC1piecewise algebraic curves of degree 2 over type-2 triangulation to the given scattered data. By simultaneously approximating points, associated normals and tangents, and points constraints, the energy term is also considered in the method. Moreover, some examples are presented.

scholarly journals Motivic zeta functions for curve singularities

2010 ◽  
Vol 198 ◽  
pp. 47-75 ◽  
Author(s):  
J. J. Moyano-Fernández ◽  
W. A. Zúňiga-Galindo
Keyword(s):  
Functional Equation ◽  
Zeta Function ◽  
Local Ring ◽  
Algebraic Curve ◽  
Algebraic Curves ◽  
Zeta Functions ◽  
Poincaré Series ◽  
Poincare Series ◽  
Motivic Zeta Functions ◽  
Curve Singularities

AbstractLet X be a complete, geometrically irreducible, singular, algebraic curve defined over a field of characteristic p big enough. Given a local ring Op,x at a rational singular point P of X, we attached a universal zeta function which is a rational function and admits a functional equation if Op,x is Gorenstein. This universal zeta function specializes to other known zeta functions and Poincaré series attached to singular points of algebraic curves. In particular, for the local ring attached to a complex analytic function in two variables, our universal zeta function specializes to the generalized Poincaré series introduced by Campillo, Delgado, and Gusein-Zade.

scholarly journals Inverse problems for invariant algebraic curves: explicit computations

2009 ◽  
Vol 139 (2) ◽  
pp. 287-302 ◽  
Author(s):  
Colin Christopher ◽  
Jaume Llibre ◽  
Chara Pantazi ◽  
Sebastian Walcher
Keyword(s):  
Inverse Problems ◽  
Algebraic Curve ◽  
Affine Plane ◽  
Vector Fields ◽  
Algebraic Curves ◽  
General Setting ◽  
Polynomial Vector ◽  
Polynomial Vector Fields ◽  
Algorithmic Approach ◽  
Invariant Algebraic Curves

Given an algebraic curve in the complex affine plane, we describe how to determine all planar polynomial vector fields which leave this curve invariant. If all (finite) singular points of the curve are non-degenerate, we give an explicit expression for these vector fields. In the general setting we provide an algorithmic approach, and as an alternative we discuss sigma processes.

AN ALGORITHM FOR FINDING INVARIANT ALGEBRAIC CURVES OF A GIVEN DEGREE FOR POLYNOMIAL PLANAR VECTOR FIELDS

2005 ◽  
Vol 15 (03) ◽  
pp. 1033-1044 ◽  
Author(s):  
GRZEGORZ ŚWIRSZCZ
Keyword(s):  
Differential Equations ◽  
Algebraic Curve ◽  
Vector Fields ◽  
Algebraic Curves ◽  
Polynomial System ◽  
Polynomial Systems ◽  
Invariant Algebraic Curves ◽  
Invariant Algebraic Curve ◽  
Planar Vector Fields ◽  
Planar Vector

Given a system of two autonomous ordinary differential equations whose right-hand sides are polynomials, it is very hard to tell if any nonsingular trajectories of the system are contained in algebraic curves. We present an effective method of deciding whether a given system has an invariant algebraic curve of a given degree. The method also allows the construction of examples of polynomial systems with invariant algebraic curves of a given degree. We present the first known example of a degree 6 algebraic saddle-loop for polynomial system of degree 2, which has been found using the described method. We also present some new examples of invariant algebraic curves of degrees 4 and 5 with an interesting geometry.

scholarly journals Motivic zeta functions for curve singularities

2010 ◽  
Vol 198 ◽  
pp. 47-75
Author(s):  
J. J. Moyano-Fernández ◽  
W. A. Zúňiga-Galindo
Keyword(s):  
Functional Equation ◽  
Zeta Function ◽  
Local Ring ◽  
Algebraic Curve ◽  
Algebraic Curves ◽  
Zeta Functions ◽  
Poincaré Series ◽  
Poincare Series ◽  
Motivic Zeta Functions ◽  
Curve Singularities

AbstractLetXbe a complete, geometrically irreducible, singular, algebraic curve defined over a field of characteristicpbig enough. Given a local ringOp,x at a rational singular pointPofX, we attached a universal zeta function which is a rational function and admits a functional equation ifOp,x is Gorenstein. This universal zeta function specializes to other known zeta functions and Poincaré series attached to singular points of algebraic curves. In particular, for the local ring attached to a complex analytic function in two variables, our universal zeta function specializes to the generalized Poincaré series introduced by Campillo, Delgado, and Gusein-Zade.

scholarly journals On the Steinitz module and capitulation of ideals

2000 ◽  
Vol 160 ◽  
pp. 1-15
Author(s):  
Chandrashekhar Khare ◽  
Dipendra Prasad
Keyword(s):  
Number Field ◽  
Algebraic Curve ◽  
Class Group ◽  
Algebraic Curves ◽  
Finite Extension ◽  
Number Fields ◽  
Line Bundles ◽  
Exterior Power ◽  
Ring Of Integers ◽  
And Function

AbstractLet L be a finite extension of a number field K with ring of integers and respectively. One can consider as a projective module over . The highest exterior power of as an module gives an element of the class group of , called the Steinitz module. These considerations work also for algebraic curves where we prove that for a finite unramified cover Y of an algebraic curve X, the Steinitz module as an element of the Picard group of X is the sum of the line bundles on X which become trivial when pulled back to Y. We give some examples to show that this kind of result is not true for number fields. We also make some remarks on the capitulation problem for both number field and function fields. (An ideal in is said to capitulate in L if its extension to is a principal ideal.)

About the Tasks of Descriptive Geometry With Imaginary Solutions

2015 ◽  
Vol 3 (2) ◽  
pp. 3-8 ◽  
Author(s):  
Иванов ◽  
G. Ivanov ◽  
Дмитриева ◽  
I. Dmitrieva
Keyword(s):  
Field Theory ◽  
Algebraic Curve ◽  
Complex Space ◽  
Algebraic Curves ◽  
Descriptive Geometry ◽  
Special Cases ◽  
Common Plane ◽  
Plane Algebraic Curve ◽  
Methodological Problems ◽  
Mathematical Interpretation

The article is devoted to the discussion of the scientific methodological problems of presentation tasks of descriptive geometry along with having real and imaginary solutions. Examples of such problems are given, graphics solutions who give the wrong answers. As a consequence they resulted in some the textbooks on descriptive geometry to the emergence false claims type “ the curve degenerates to a point”, “a torus is a surface of the second order”, “conical and cylindrical surfaces are a special cases of the torsoboy surface in the case of degeneration of the ribs return torsoboy the surface at the point, etc.” In the article gives a correct mathematical interpretation of imaginary solutions the tasks by considering of examples an the determine the order and class of plane algebraic curve, the isolated point touch, of the line of intersection of surfaces of the second order with a common plane of symmetry. To obtain a mathematically valid answers the conclusion about the need for a combination of graphical and analytical solutions. This approach meets the requirements of the GEF on ensure as intrasubject discussed in this publication, and so interdisciplinary competencies. The latter have a broad outlet of descriptive geometry in complex space in the theory of algebraic curves and surfaces, kremenovic transformations, field theory, etc.

Some enumerative formulae for algebraic curves

1958 ◽  
Vol 54 (4) ◽  
pp. 399-416 ◽  
Author(s):  
I. G. Macdonald
Keyword(s):  
Equivalence Relation ◽  
Algebraic Curve ◽  
Algebraic Curves ◽  
Linear Series ◽  
Set Of Points ◽  
Algebraic Correspondence

This paper is in two parts. In Part I we are concerned with one or more linear series on an algebraic curve; we consider a set of points on the curve which are contained with assigned multiplicities in a set of each of the linear series and, by persistent use of Severi's equivalence relation for the united points of an algebraic correspondence with valency, we derive formulae for the number of such sets of points when the constants involved are such as to make this number finite. All this is essentially a generalization of the formula for the number of points in the Jacobian set of a linear series of freedom 1, and the main result is Theorem 3.

scholarly journals Automorphism Groups of Symmetric and Pseudo-real Riemann Surfaces

2021 ◽  
Vol 18 (5) ◽  
Author(s):  
Ewa Tyszkowska
Keyword(s):  
Riemann Surfaces ◽  
Algebraic Curve ◽  
Algebraic Curves ◽  
Automorphism Groups ◽  
Real Riemann Surfaces ◽  
Compact Riemann Surfaces

AbstractThe category of smooth, irreducible, projective, complex algebraic curves is equivalent to the category of compact Riemann surfaces. We study automorphism groups of Riemann surfaces which are equivalent to complex algebraic curves with real moduli. A complex algebraic curve C has real moduli when the corresponding surface $$X_C$$ X C admits an anti-conformal automorphism. If no such an automorphism is an involution (symmetry), then the surface $$X_C$$ X C is called pseudo-real and the curve C is isomorphic to its conjugate, but is not definable over reals. Otherwise, the surface $$X_C$$ X C is called symmetric and the curve C is real.

Regulous Functions over Real Closed Fields

2021 ◽  
Author(s):  
Wojciech Kucharz ◽  
Krzysztof Kurdyka
Keyword(s):  
Algebraic Variety ◽  
Open Subset ◽  
Algebraic Curve ◽  
Algebraic Curves ◽  
Rational Points ◽  
Real Closed Field ◽  
Closed Field ◽  
Rational Representation ◽  
New Methods ◽  
Closed Fields

Abstract Let $X$ be a quasi-projective algebraic variety over a real closed field $R$, and let $f \colon U \to R$ be a function defined on an open subset $U$ of the set $X(R)$ of $R$-rational points of $X$. Assume that either the function $f$ is locally semialgebraic or the field $R$ is uncountable. If for every irreducible algebraic curve $C \subset X$ the restriction $f|_{U \cap C}$ is continuous and admits a rational representation, then $f$ is continuous and admits a rational representation. There are also suitable versions of this theorem with algebraic curves replaced by algebraic arcs. Heretofore, results of such a type have been known only for $R={\mathbb{R}}$. The transition from ${\mathbb{R}}$ to $R$ is not automatic at all and requires new methods.

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