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Showing 1 to 13 of 13 for “"Algebraic curves"”.
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Algebraic Curves Over Supersimple Fields
In this work we are concerned with algebraic curves over supersimple fields. First it is proved that an elliptic curve defined over a supersimple field K whose j-invariant is s-generic over empty has a rational point over K which is s-generic over the set of parameters used to define the curve. …
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Applications of Algebraic Curves to Cryptography
Secondly, we use algebraic functions with two poles to obtain efficient secret sharing schemes. We present a method to find the lower bounds for the minimum distance of geometric codes. We apply this to the two-point codes on a Hermitian function field. The lower bounds turn out to be sharp and …
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Improved Bounds for Codes and Secret Sharing Schemes from Algebraic Curves
The main goal of this work is to improve algebraic geometric/number theoretic constructions of error-correcting codes and secret sharing schemes. For both objects we define parameters that indicate their effectiveness in applications. We explore infeasibility bounds, showing that objects with …
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Improving the Stability of the Recovery of Algebraic Curves via Bernstein Basis Polynomials and Neural Networks
… that we consider are characteristic functions of algebraic shapes, that is, interiors of zero sets of bivariate polynomials, and we assume that we only know a finite set of samples of these images. A solution to this problem can be formulated in terms of a system of linear equations of moments. …
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The Arithmetic of Elliptic and Hyperelliptic Curves with Applications to Pairing-Based Cryptography
… giving an extensive treatment of the theory of algebraic curves and their connection to the theory of algebraic function fields of one variable, the thesis concentrates on pairings (such as the Tate pairing) defined on groups related to a given absolutely irreducible non-singular curve over a …
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Elliptic Curves Over Finite Fields
… a self-contained introduction to elliptic curves accessible to advanced undergraduates and graduate students in mathematics, with emphasis on the the theory of elliptic curves over finite fields. In Chapter 1, affine and projective planes are introduced. Chapter 2 introduces the theory of …
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The Absolute Galois Group of the Rationals, Grothendieck's Dessin D'Enfants, and Galois Invariants
… equivalence between the category of irreducible algebraic curves over the algebraic closure of the rationals and the category of finite covers of the Riemann sphere ramified at three points. We observe that the latter is also equivalent to the category of A. Grothendieck's dessins d'enfants: …
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Tropical Intersection Theory on Moduli Stack of Curve Coverings
… covers (i.e., coverings of twisted tropical curves). We define the tropical intersection theory on $\mathfrac{M}_\eta^\text{trop}$ and show that the tropical intersection theory agrees with the intersection theory on the moduli stack $\bar{\mathfrac{M}}_\eta$ of \'{e}tale covers (i.e., …
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Covers and invariants of Deligne-Lusztig curves
… one or more of the Hermitian, Suzuki, and Ree curves, which are three families of algebraic curves over finite fields which have pronounced arithmetic and geometric properties. In the first part, we use a ray class field construction to produce covers of each of these three families of curves …
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Rank 3 permuation characters and maximal subgroups
… an application to the study of minimal genus of algebraic curves which admit group actions.
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Log geometry and extremal contractions
… (in short, MMP) aims at classifying projective algebraic varieties from a birational point of view. That means that starting from a projective algebraic variety X, [Delta] it is allowed to change the variety under scrutiny as long as its field of rational functions remains the same. In this …
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Chebyshev-like polynomials, conic distribution of roots, and continued fractions
… second kind lie very close to certain high order algebraic curves that are ``nearly'' ellipses. Next, we examine roots of linear combinations of Chebyshev polynomials. Here we use continued fractions to give estimates for the roots that do not lie in the interval $(-1,1)$. We then show the …
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Graded Lie Algebras, Compactified Jacobians and Arithmetic Statistics
… laced Dynkin diagram gives rise to a family of curves over Q and a coregular representation, using deformations of simple singularities and Vinberg theory respectively. Thorne has conjectured and partially proven a strong link between the arithmetic of these curves and the rational orbits of …