Showing papers on "Field (mathematics) published in 2015"

Algebraic Coding Theory

Abstract: This is the revised edition of Berlekamp's famous book, "Algebraic Coding Theory," originally published in 1968, wherein he introduced several algorithms which have subsequently dominated engineering practice in this field. One of these is an algorithm for decoding Reed-Solomon and Bose–Chaudhuri–Hocquenghem codes that subsequently became known as the Berlekamp–Massey Algorithm. Another is the Berlekamp algorithm for factoring polynomials over finite fields, whose later extensions and embellishments became widely used in symbolic manipulation systems. Other novel algorithms improved the basic methods for doing various arithmetic operations in finite fields of characteristic two. Other major research contributions in this book included a new class of Lee metric codes, and precise asymptotic results on the number of information symbols in long binary BCH codes.Selected chapters of the book became a standard graduate textbook.Both practicing engineers and scholars will find this book to be of great value.

An Introduction to Polynomial and Semi-Algebraic Optimization

Abstract: This is the first comprehensive introduction to the powerful moment approach for solving global optimization problems (and some related problems) described by polynomials (and even semi-algebraic functions). In particular, the author explains how to use relatively recent results from real algebraic geometry to provide a systematic numerical scheme for computing the optimal value and global minimizers. Indeed, among other things, powerful positivity certificates from real algebraic geometry allow one to define an appropriate hierarchy of semidefinite (SOS) relaxations or LP relaxations whose optimal values converge to the global minimum. Several extensions to related optimization problems are also described. Graduate students, engineers and researchers entering the field can use this book to understand, experiment with and master this new approach through the simple worked examples provided.

Algebraic Coding Theory: Revised Edition

Abstract: This is the revised edition of Berlekamp's famous book, "Algebraic Coding Theory", originally published in 1968, wherein he introduced several algorithms which have subsequently dominated engineering practice in this field. One of these is an algorithm for decoding Reed-Solomon and BoseChaudhuriHocquenghem codes that subsequently became known as the BerlekampMassey Algorithm. Another is the Berlekamp algorithm for factoring polynomials over finite fields, whose later extensions and embellishments became widely used in symbolic manipulation systems. Other novel algorithms improved the basic methods for doing various arithmetic operations in finite fields of characteristic two. Other major research contributions in this book included a new class of Lee metric codes, and precise asymptotic results on the number of information symbols in long binary BCH codes. Selected chapters of the book became a standard graduate textbook. Both practicing engineers and scholars will find this book to be of great value. Readership: Researchers in coding theory and cryptography, algebra and number theory, and software engineering.

Unit Equations in Diophantine Number Theory

Abstract: Diophantine number theory is an active area that has seen tremendous growth over the past century, and in this theory unit equations play a central role. This comprehensive treatment is the first volume devoted to these equations. The authors gather together all the most important results and look at many different aspects, including effective results on unit equations over number fields, estimates on the number of solutions, analogues for function fields and effective results for unit equations over finitely generated domains. They also present a variety of applications. Introductory chapters provide the necessary background in algebraic number theory and function field theory, as well as an account of the required tools from Diophantine approximation and transcendence theory. This makes the book suitable for young researchers as well as experts who are looking for an up-to-date overview of the field.

Cyclic LRC codes and their subfield subcodes

Abstract: We consider linear cyclic codes with the locality property, or locally recoverable codes (LRC codes). A family of LRC codes that generalizes the classical construction of Reed-Solomon codes was constructed in a recent paper by I. Tamo and A. Barg (IEEE Trans. IT, no. 8, 2014). In this paper we focus on the optimal cyclic codes that arise from the general construction. We give a characterization of these codes in terms of their zeros, and observe that there are many equivalent ways of constructing optimal cyclic LRC codes over a given field. We also study subfield subcodes of cyclic LRC codes (BCH-like LRC codes) and establish several results about their locality and minimum distance.

Maximum of a log-correlated Gaussian field

Abstract: Nous etudions le maximum d’un champ Gaussien sur $[0,1]^{\mathtt{d}}$ ($\mathtt{d}\geq1$) dont les correlations decroissent logarithmiquement avec la distance. Kahane (Ann. Sci. Math. Quebec 9 (1985) 105–150) a introduit ce modele pour construire mathematiquement le chaos Gaussien multiplicatif dans le cas sous-critique. Duplantier, Rhodes, Sheffield et Vargas (Critical Gaussian multiplicative chaos: Convergence of the derivative martingale (2012) Preprint, Renormalization of critical Gaussian multiplicative chaos and KPZ formula (2012) Preprint) ont etendu cette construction au cas critique et ont etabli la formule KPZ. De plus, dans (Critical Gaussian multiplicative chaos: Convergence of the derivative martingale (2012) Preprint), ils fournissent plusieurs conjectures sur le cas sur-critique ainsi que sur le maximum de ce champ Gaussien. Dans ce papier nous etablissons la convergence en loi du maximum et montrons que loi limite est une variable aleatoire de Gumbel convoluee avec la limite de la martingale derivee, resolvant ainsi la Conjecture 12 de (Critical Gaussian multiplicative chaos: Convergence of the derivative martingale (2012) Preprint).

Field reduction and linear sets in finite geometry

Abstract: Based on the simple and well understood concept of subfields in a finite field, the technique called 'field reduction' has proved to be a very useful and powerful tool in finite geometry. In this paper we elaborate on this technique. Field reduction for projective and polar spaces is formalised and the links with Desarguesian spreads and linear sets are explained in detail. Recent results and some fundamental questions about linear sets and scattered spaces are studied. The relevance of field reduction is illustrated by discussing applications to blocking sets and semifields.

On exact correlation functions in SU(N) ${\cal N} = 2$ superconformal QCD

Abstract: We consider the exact coupling constant dependence of extremal correlation functions of ${\cal N} = 2$ chiral primary operators in 4d ${\cal N} = 2$ superconformal gauge theories with gauge group SU(N) and N_f=2N massless fundamental hypermultiplets. The 2- and 3-point functions, viewed as functions of the exactly marginal coupling constant and theta angle, obey the tt* equations. In the case at hand, the tt* equations form a set of complicated non-linear coupled matrix equations. We point out that there is an ad hoc self-consistent ansatz that reduces this set of partial differential equations to a sequence of decoupled semi-infinite Toda chains, similar to the one encountered previously in the special case of SU(2) gauge group. This ansatz requires a surprising new non-renormalization theorem in ${\cal N} = 2$ superconformal field theories. We derive a general 3-loop perturbative formula for 2- and 3-point functions in the ${\cal N} = 2$ chiral ring of the SU(N) theory, and in all explicitly computed examples we find agreement with the tt* equations, as well as the above-mentioned ansatz. This is suggestive evidence for an interesting non-perturbative conjecture about the structure of the ${\cal N} = 2$ chiral ring in this class of theories. We discuss several implications of this conjecture. For example, it implies that the holonomy of the vector bundles of chiral primaries over the superconformal manifold is reducible. It also implies that a specific subset of extremal correlation functions can be computed in the SU(N) theory using information solely from the S^4 partition function of the theory obtained by supersymmetric localization.

A quadratic refinement of the Grothendieck–Lefschetz–Verdier trace formula

Abstract: We prove a trace formula in stable motivic homotopy theory over a general base scheme, equating the trace of an endomorphism of a smooth proper scheme with the “Euler characteristic integral” of a certain cohomotopy class over its scheme of fixed points. When the base is a field and the fixed points are etale, we compute this integral in terms of Morel’s identification of the ring of endomorphisms of the motivic sphere spectrum with the Grothendieck‐Witt ring. In particular, we show that the Euler characteristic of an etale algebra corresponds to the class of its trace form in the Grothendieck‐Witt ring. 14F42; 47H10, 11E81

Universal Gauss-Thakur sums and L-series ∗†

Abstract: We prove that the maximal abelian extension tamely ramified at infinity of the rational function field over \(\mathbb {F}_q\) is generated by the values at the points in the algebraic closure of \(\mathbb {F}_q\) of the higher derivatives of the so-called Anderson and Thakur function \(\omega .\) We deduce a similar property for the special values of the higher derivatives of a new kind of \(L\)-series introduced by the second author.

Algebraic Groups and Compact Generation of their Derived Categories of Representations

Abstract: Let k be a field. We characterize the group schemes G over k, not necessarily affine, such that D-qc (B(k)G) is compactly generated. We also describe the algebraic stacks that have finite cohomological dimension in terms of their stabilizer groups.

Good reduction criterion for K3 surfaces

Abstract: We prove a Neron–Ogg–Shafarevich type criterion for good reduction of K3 surfaces, which states that a K3 surface over a complete discrete valuation field has potential good reduction if its \(l\)-adic cohomology group is unramified. We also prove a \(p\)-adic version of the criterion. (These are analogues of the criteria for good reduction of abelian varieties.) The model of the surface will be in general not a scheme but an algebraic space. As a corollary of the criterion we obtain the surjectivity of the period map of K3 surfaces in positive characteristic.

Improving NFS for the discrete logarithm problem in non-prime finite fields

Abstract: The aim of this work is to investigate the hardness of the discrete logarithm problem in fields GF(\(p^n\)) where \(n\) is a small integer greater than \(1\). Though less studied than the small characteristic case or the prime field case, the difficulty of this problem is at the heart of security evaluations for torus-based and pairing-based cryptography. The best known method for solving this problem is the Number Field Sieve (NFS). A key ingredient in this algorithm is the ability to find good polynomials that define the extension fields used in NFS. We design two new methods for this task, modifying the asymptotic complexity and paving the way for record-breaking computations. We exemplify these results with the computation of discrete logarithms over a field GF(\(p^2\)) whose cardinality is 180 digits (595 bits) long.

Generalized ADE classification of topological boundaries and anyon condensation

Abstract: In this paper we would like to demonstrate how the known, physically-motivat-ed rules of anyon condensation proposed by Bais et al. can be recovered by the mathematics of twist-free commutative separable Frobenius algebra (CSFA). In some simple cases, those physical rules are also sufficient conditions defining a twist-free CSFA. This allows us to make use of the generalized ADE classification of CSFA’s and modular invariants to classify anyon condensation, characterize the topological boundaries between topological field theories and thus describe all gapped domain walls and gapped boundaries of a large class of topological orders. In fact, this classification is equivalent to the classification we proposed in ref. [1].

A combinatorial Li–Yau inequality and rational points on curves

Abstract: We present a method to control gonality of nonarchimedean curves based on graph theory. Let $$k$$ denote a complete nonarchimedean valued field. We first prove a lower bound for the gonality of a curve over the algebraic closure of $$k$$ in terms of the minimal degree of a class of graph maps, namely: one should minimize over all so-called finite harmonic graph morphisms to trees, that originate from any refinement of the dual graph of the stable model of the curve. Next comes our main result: we prove a lower bound for the degree of such a graph morphism in terms of the first eigenvalue of the Laplacian and some “volume” of the original graph; this can be seen as a substitute for graphs of the Li–Yau inequality from differential geometry, although we also prove that the strict analogue of the original inequality fails for general graphs. Finally, we apply the results to give a lower bound for the gonality of arbitrary Drinfeld modular curves over finite fields and for general congruence subgroups $$\varGamma $$ of $$\varGamma (1)$$ that is linear in the index $$[\varGamma (1):\varGamma ]$$ , with a constant that only depends on the residue field degree and the degree of the chosen “infinite” place. This is a function field analogue of a theorem of Abramovich for classical modular curves. We present applications to uniform boundedness of torsion of rank two Drinfeld modules that improve upon existing results, and to lower bounds on the modular degree of certain elliptic curves over function fields that solve a problem of Papikian.

$(0, 4)$ dualities

Abstract: We study a class of two-dimensional ${\cal N}=(0, 4)$ quiver gauge theories that flow to superconformal field theories. We find dualities for the superconformal field theories similar to the 4d ${\cal N}=2$ theories of class ${\cal S}$, labelled by a Riemann surface ${\cal C}$. The dual descriptions arise from various pair-of-pants decompositions, that involves an analog of the $T_N$ theory. Especially, we find the superconformal index of such theories can be written in terms of a topological field theory on ${\cal C}$. We interpret this class of SCFTs as the ones coming from compactifying 6d ${\cal N}=(2, 0)$ theory on $\mathbb{CP}^1 \times {\cal C}$

Interacting Hopf Algebras: the theory of linear systems

Abstract: We present by generators and equations the algebraic theory IH whose free model is the category oflinear subspaces over a field k. Terms of IH are string diagrams which, for different choices of k, expressdifferent kinds of networks and graphical formalisms used by scientists in various fields, such as quantumcircuits, electrical circuits and Petri nets. The equations of IH arise by distributive laws between Hopfalgebras - from which the name interacting Hopf algebras. The characterisation in terms of subspacesallows to think of IH as a string diagrammatic syntax for linear algebra: linear maps, spaces and theirtransformations are all faithfully represented in the graphical language, resulting in an alternative, ofteninsightful perspective on the subject matter. As main application, we use IH to axiomatise a formalsemantics of signal processing circuits, for which we study full abstraction and realisability. Our analysissuggests a reflection about the role of causality in the semantics of computing devices.

Constructible sheaves are holonomic

Abstract: We show that for any constructible sheaf F on a smooth algebraic variety X over a field of arbitrary characteristic its singular support SS(F) is equidimensional of dimension dim X. Here SS(F) is the minimal closed subset of the cotangent bundle of X such that every (local) function on X with df(X) disjoint from SS(F) is locally acyclic relative to F.

Relative) dynamical degrees of rational maps over an algebraic closed field

Abstract: The main purpose of this paper is to define dynamical degrees for rational maps over an algebraic closed field of characteristic zero and prove some basic properties (such as log-concavity) and give some applications. We also define relative dynamical degrees and prove a "product formula" for dynamical degrees of semi-conjugate rational maps in the algebraic setting. The main tools are the Chow's moving lemma and a formula for the degree of the cone over a subvariety of $\mathbb{P}^N$. The proofs of these results are valid as long as resolution of singularities are available (or more generally if appropriate birational models of the maps under consideration are available). This observation is applied for the cases of surfaces and threefolds over a field of positive characteristic.

On a generalization of close-to-convex functions

Abstract: A motivation comes from {\em M. Ismail and et al.: A generalization of starlike functions, Complex Variables Theory Appl., 14 (1990), 77--84} to study a generalization of close-to-convex functions by means of a $q$-analog of a difference operator acting on analytic functions in the unit disk $\mathbb{D}=\{z\in \mathbb{C}:\,|z|<1\}$. We use the terminology {\em $q$-close-to-convex functions} for the $q$-analog of close-to-convex functions. The $q$-theory has wide applications in special functions and quantum physics which makes the study interesting and pertinent in this field. In this paper, we obtain some interesting results concerning conditions on the coefficients of power series of functions analytic in the unit disk which ensure that they generate functions in the $q$-close-to-convex family. As a result we find certain dilogarithm functions that are contained in this family. Secondly, we also study the famous Bieberbach conjecture problem on coefficients of analytic $q$-close-to-convex functions. This produces several power series of analytic functions convergent to basic hypergeometric functions.

On the Extended W-Algebra of Type 2 at Positive Rational Level

Abstract: The extended W-algebra of type sl2 at positive rational level, denoted by Mp+,p− , is a vertex operator algebra (VOA) that was originally proposed in [16]. This VOA is an extension of the minimal model VOA and plays the role of symmetry algebra for certain logarithmic conformal field theories. We give a construction of Mp+,p− in terms of screening operators and use this construction to prove that Mp+,p− satisfies Zhu's c2-cofiniteness condition, calculate the structure of the zero mode algebra (also known as Zhu's algebra), and classify all simple Mp+,p− -modules.

Some structure theorems for algebraic groups

Abstract: These are extended notes of a course given at Tulane University for the 2015 Clifford Lectures. Their aim is to present structure results for group schemes of finite type over a field, with applications to Picard varieties and automorphism groups.

Topology driven modeling: the IS metaphor

Abstract: In order to define a new method for analyzing the immune system within the realm of Big Data, we bear on the metaphor provided by an extension of Parisi's model, based on a mean field approach. The novelty is the multilinearity of the couplings in the configurational variables. This peculiarity allows us to compare the partition function $$Z$$Z with a particular functor of topological field theory--the generating function of the Betti numbers of the state manifold of the system--which contains the same global information of the system configurations and of the data set representing them. The comparison between the Betti numbers of the model and the real Betti numbers obtained from the topological analysis of phenomenological data, is expected to discover hidden n-ary relations among idiotypes and anti-idiotypes. The data topological analysis will select global features, reducible neither to a mere subgraph nor to a metric or vector space. How the immune system reacts, how it evolves, how it responds to stimuli is the result of an interaction that took place among many entities constrained in specific configurations which are relational. Within this metaphor, the proposed method turns out to be a global topological application of the S[B] paradigm for modeling complex systems.

Higher derivative extensions of 3d Chern–Simons models: conservation laws and stability

Abstract: We consider the class of higher derivative 3d vector field models with the field equation operator being a polynomial of the Chern–Simons operator. For the nth-order theory of this type, we provide a general recipe for constructing n-parameter family of conserved second rank tensors. The family includes the canonical energy-momentum tensor, which is unbounded, while there are bounded conserved tensors that provide classical stability of the system for certain combinations of the parameters in the Lagrangian. We also demonstrate the examples of consistent interactions which are compatible with the requirement of stability.

Sums of divisor functions in $F_{q}[t]$ and matrix integrals

Abstract: We study the mean square of sums of the $k$th divisor function $d_k(n)$ over short intervals and arithmetic progressions for the rational function field over a finite field of $q$ elements. In the limit as $q\rightarrow\infty$ we establish a relationship with a matrix integral over the unitary group. Evaluating this integral enables us to compute the mean square of the sums of $d_k(n)$ in terms of a lattice point count. This lattice point count can in turn be calculated in terms of certain polynomials, which we analyse. Our results suggest general conjectures for the corresponding classical problems over the integers, which agree with the few cases where the answer is known.

Symbol length in the Brauer group of a field

Abstract: We bound the symbol length of elements in the Brauer group of a field K containing a Cm field (for example any field containing an algebraically closed field or a finite field), and solve the local exponent-index problem for a Cm field F . In particular, for a Cm field F , we show that every F central simple algebra of exponent p is similar to the tensor product of at most len(p, F ) ≤ t(pm−1−1) symbol algebras of degree p. We then use this bound on the symbol length to show that the index of such algebras is bounded by (p) −1), which in turn gives a bound for any algebra of exponent n via the primary decomposition. Finally for a field K containing a Cm field F , we show that every F central simple algebra of exponent p and degree p is similar to the tensor product of at most len(p, p,K) ≤ len(p, L) symbol algebras of degree p, where L is a Cm+edL(A)+ps−t−1 field.

Asymptotic entanglement transformation between W and GHZ states

Abstract: We investigate entanglement transformations with stochastic local operations and classical communication in an asymptotic setting using the concepts of degeneration and border rank of tensors from algebraic complexity theory. Results well-known in that field imply that GHZ states can be transformed into W states at rate 1 for any number of parties. As a generalization, we find that the asymptotic conversion rate from GHZ states to Dicke states is bounded as the number of subsystems increases and the number of excitations is fixed. By generalizing constructions of Coppersmith and Winograd and by using monotones introduced by Strassen, we also compute the conversion rate from W to GHZ states.