Showing papers on "Algebraic number published in 2010"

The Arithmetic of Dynamical Systems

Abstract: * Provides an entry for graduate students into an active field of research * Each chapter includes exercises, examples, and figures * Will become a standard reference for researchers in the field * Contains descriptions of many known results and conjectures, together with an extensive bibliography This book provides an introduction to the relatively new discipline of arithmetic dynamics. Whereas classical discrete dynamics is the study of iteration of self-maps of the complex plane or real line, arithmetic dynamics is the study of the number-theoretic properties of rational and algebraic points under repeated application of a polynomial or rational function. A principal theme of arithmetic dynamics is that many of the fundamental problems in the theory of Diophantine equations have dynamical analogs. As is typical in any subject combining Diophantine problems and geometry, a fundamental goal is to describe arithmetic properties, at least qualitatively, in terms of underlying geometric structures. Key features: - Provides an entry for graduate students into an active field of research - Provides a standard reference source for researchers - Includes numerous exercises and examples - Contains a description of many known results and conjectures, as well as an extensive glossary, bibliography, and index This graduate-level text assumes familiarity with basic algebraic number theory. Other topics, such as basic algebraic geometry, elliptic curves, nonarchimedean analysis, and the theory of Diophantine approximation, are introduced and referenced as needed. Mathematicians and graduate students will find this text to be an excellent reference.

Division Algebras:: Octonions Quaternions Complex Numbers and the Algebraic Design of Physics

Abstract: I. Underpinnings. II. Division Algebra Alone. III. Tensor Algebras. IV. Connecting to Physics. V. Spontaneous Symmetry Breaking. VI. 10 Dimensions. VII. Doorways. VIII. Corridors. Appendices. Bibliography. Index.

Symmetric informationally complete positive-operator-valued measures: A new computer study

Abstract: We report on a new computer study of the existence of d2 equiangular lines in d complex dimensions. Such maximal complex projective codes are conjectured to exist in all finite dimensions and are the underlying mathematical objects defining symmetric informationally complete measurements in quantum theory. We provide numerical solutions in all dimensions d≤67 and, moreover, a putatively complete list of Weyl–Heisenberg covariant solutions for d≤50. A symmetry analysis of this list leads to new algebraic solutions in dimensions d=24, 35, and 48, which are given together with algebraic solutions for d=4,…,15, and 19.

Noncommutative Rational Series with Applications

Abstract: The algebraic theory of automata was created by Schutzenberger and Chomsky over 50 years ago and there has since been a great deal of development. Classical work on the theory to noncommutative power series has been augmented more recently to areas such as representation theory, combinatorial mathematics and theoretical computer science. This book presents to an audience of graduate students and researchers a modern account of the subject and its applications. The algebraic approach allows the theory to be developed in a general form of wide applicability. For example, number-theoretic results can now be more fully explored, in addition to applications in automata theory, codes and non-commutative algebra. Much material, for example, Schutzenberger's theorem on polynomially bounded rational series, appears here for the first time in book form. This is an excellent resource and reference for all those working in algebra, theoretical computer science and their areas of overlap.

Character Sums with Exponential Functions and their Applications

Abstract: Part I. Preliminaries: 1. Introduction 2. Notation and auxiliary results Part II. Bounds of Character Sums: 3. Bounds of long character sums 4. Bounds of short character sums 5. Bounds of character sums for almost all moduli 6. Bounds of Gaussian sums Part III. Multiplicative Translations of Sets: 7. Multiplicative translations of subgroups of F*p 8. Multiplicative translations of arbitrary sets modulo p Part IV. Applications to Algebraic Number Fields: 9 Representatives of residue classes 10. Cyclotomic fields and Gaussian periods Part V. Applications to Pseudo-random Number Generators: 11. Prediction of pseudo-random number generators 12. Congruential pseudo-random number generators Part VI. Applications to Finite Fields: 13. Small mth roots modulo p 14. Supersingular hyperelliptic curves 15. Distribution of powers of primitive roots 16. Difference sets in Vp 17. Dimension of BCH codes 18. An enumeration problem in finite fields.

Some new classes of complex symmetric operators

Abstract: We say that an operator T E B(H) is complex symmetric if there exists a conjugate-linear, isometric involution C: ℌ→ℌ so that T = CT * C. We prove that binormal operators, operators that are algebraic of degree two (including all idempotents), and large classes of rank-one perturbations of normal operators are complex symmetric. From an abstract viewpoint, these results explain why the compressed shift and Volterra integration operator are complex symmetric. Finally, we attempt to describe all complex symmetric partial isometries, obtaining the sharpest possible statement given only the data (dim ker T, dim ker T*).

A universal characterization of higher algebraic K-theory

Abstract: In this paper we establish a universal characterization of higher algebraic K-theory in the setting of small stable infinity categories. Specifically, we prove that connective algebraic K-theory is the universal additive invariant, i.e., the universal functor with values in spectra which inverts Morita equivalences, preserves filtered colimits, and satisfies Waldhausen's additivity theorem. Similarly, we prove that non-connective algebraic K-theory is the universal localizing invariant, i.e., the universal functor that moreover satisfies the "Thomason-Trobaugh-Neeman" localization theorem. To prove these results, we construct and study two stable infinity categories of "noncommutative motives"; one associated to additivity and another to localization. In these stable infinity categories, Waldhausen's S. construction corresponds to the suspension functor and connective and non-connective algebraic K-theory spectra become corepresentable by the noncommutative motive of the sphere spectrum. In particular, the algebraic K-theory of every scheme, stack, and ring spectrum can be recovered from these categories of noncommutative motives. In order to work with these categories of noncommutative motives, we establish comparison theorems between the category of spectral categories localized at the Morita equivalences and the category of small idempotent-complete stable infinity categories. We also explain in detail the comparison between the infinity categorical version of Waldhausen K-theory and the classical definition. As an application of our theory, we obtain a complete classification of the natural transformations from higher algebraic K-theory to topological Hochschild homology (THH) and topological cyclic homology (TC). Notably, we obtain an elegant conceptual description of the cyclotomic trace map.

MetiTarski: An Automatic Theorem Prover for Real-Valued Special Functions

Abstract: Many theorems involving special functions such as ln, exp and sin can be proved automatically by MetiTarski: a resolution theorem prover modified to call a decision procedure for the theory of real closed fields. Special functions are approximated by upper and lower bounds, which are typically rational functions derived from Taylor or continued fraction expansions. The decision procedure simplifies clauses by deleting literals that are inconsistent with other algebraic facts. MetiTarski simplifies arithmetic expressions by conversion to a recursive representation, followed by flattening of nested quotients. Applications include verifying hybrid and control systems.

A Complexity Dichotomy for Partition Functions with Mixed Signs

Abstract: Partition functions, also known as homomorphism functions, form a rich family of graph invariants that contain combinatorial invariants such as the number of $k$-colorings or the number of independent sets of a graph and also the partition functions of certain “spin glass” models of statistical physics such as the Ising model. Building on earlier work by Dyer and Greenhill [Random Structures Algorithms, 17 (2000), pp. 260-289] and Bulatov and Grohe [Theoret. Comput. Sci., 348 (2005), pp. 148-186], we completely classify the computational complexity of partition functions. Our main result is a dichotomy theorem stating that every partition function is either computable in polynomial time or #P-complete. Partition functions are described by symmetric matrices with real entries, and we prove that it is decidable in polynomial time in terms of the matrix whether a given partition function is in polynomial time or #P-complete. While in general it is very complicated to give an explicit algebraic or combinatorial description of the tractable cases, for partition functions described by Hadamard matrices (these turn out to be central in our proofs) we obtain a simple algebraic tractability criterion, which says that the tractable cases are those “representable” by a quadratic polynomial over the field $\mathbb{F}_2$.

A Posteriori Error Estimates Including Algebraic Error and Stopping Criteria for Iterative Solvers

Abstract: For the finite volume discretization of a second-order elliptic model problem, we derive a posteriori error estimates which take into account an inexact solution of the associated linear algebraic system. We show that the algebraic error can be bounded by constructing an equilibrated Raviart-Thomas-Nedelec discrete vector field whose divergence is given by a proper weighting of the residual vector. Next, claiming that the discretization error and the algebraic one should be in balance, we construct stopping criteria for iterative algebraic solvers. An attention is paid, in particular, to the conjugate gradient method which minimizes the energy norm of the algebraic error. Using this convenient balance, we also prove the efficiency of our a posteriori estimates; i.e., we show that they also represent a lower bound, up to a generic constant, for the overall energy error. A local version of this result is also stated. This makes our approach suitable for adaptive mesh refinement which also takes into account the algebraic error. Numerical experiments illustrate the proposed estimates and construction of efficient stopping criteria for algebraic iterative solvers.

Computing loci of rank defects of linear matrices using Gröbner bases and applications to cryptology

Abstract: Computing loci of rank defects of linear matrices (also called the MinRank problem) is a fundamental NP-hard problem of linear algebra which has applications in Cryptology, in Error Correcting Codes and in Geometry Given a square linear matrix (ie a matrix whose entries are k-variate linear forms) of size n and an integer r, the problem is to find points such that the evaluation of the matrix has rank less than r + 1 The aim of the paper is to obtain the most efficient algorithm to solve this problem To this end, we give the theoretical and practical complexity of computing Grobner bases of two algebraic formulations of the MinRank problem Both modelings lead to structured algebraic systemsThe first modeling, proposed by Kipnis and Shamir generates bi-homogeneous equations of bi-degree (1, 1) The second one is classically obtained by the vanishing of the (r + 1)-minors of the given matrix, giving rise to a determinantal ideal In both cases, under genericity assumptions on the entries of the considered matrix, we give new bounds on the degree of regularity of the considered ideal which allows us to estimate the complexity of the whole Grobner bases computations For instance, the exact degree of regularity of the determinantal ideal formulation of a generic well-defined MinRank problem is r(n - r) + 1 We also give optimal degree bounds of the loci of rank defect which are reached under genericity assumptions; the new bounds are much lower than the standard multi-homogeneous Bezout bounds (or mixed volume of Newton polytopes)TAs a by-product, we prove that the generic MinRank problem could be solved in polynomial time in n (when n - r is fixed) as announced in a previous paper of Faugere, Levy-dit-Vehel and Perret Moreover, using the determinantal ideal formulation, these results are used to break a cryptographic challenge (which was untractable so far) and allow us to evaluate precisely the security of the cryptosystem wrt n, r and k Our practical results suggest that, up to the software state of the art, this latter formulation is more adapted in the context of Grobner bases computations

Hermitian K -theory of exact categories

Abstract: We study the theory of higher Grothendieck-Witt groups, alias algebraic hermitian K-theory, of symmetric bilinear forms in exact categories, and prove additivity, cofinality, devissage and localization theorems – preparing the ground for the theory of higher Grothendieck-Witt groups of schemes as developed in [Sch08a] and [Sch08b]. No assumption on the characteristic is being made.

From Matrix Models and quantum fields to Hurwitz space and the absolute Galois group

Abstract: We show that correlators of the hermitian one-Matrix model with a general potential can be mapped to the counting of certain triples of permutations and hence to counting of holomorphic maps from world-sheet to sphere target with three branch points on the target. This allows the use of old matrix model results to derive new explicit formulae for a class of Hurwitz numbers. Holomorphic maps with three branch points are related, by Belyi's theorem, to curves and maps defined over algebraic numbers $\bmQ$. This shows that the string theory dual of the one-matrix model at generic couplings has worldsheets defined over the algebraic numbers and a target space $ \mP^1 (\bmQ)$. The absolute Galois group $ Gal (\bmQ / \mQ) $ acts on the Feynman diagrams of the 1-matrix model, which are related to Grothendieck's Dessins d'Enfants. Correlators of multi-matrix models are mapped to the counting of triples of permutations subject to equivalences defined by subgroups of the permutation groups. This is related to colorings of the edges of the Grothendieck Dessins. The colored-edge Dessins are useful as a tool for describing some known invariants of the $ Gal (\bmQ / \mQ) $ action on Grothendieck Dessins and for defining new invariants.

Good formal structures for flat meromorphic connections, I: Surfaces

Abstract: We give a criterion under which one can obtain a good decomposition (in the sense of Malgrange) of a formal flat connection on a complex analytic or algebraic variety of arbitrary dimension. The criterion is stated in terms of the spectral behavior of differential operators and generalizes Robba's construction of the Hukuhara-Levelt-Turrittin decomposition in the one-dimensional case. As an application, we prove the existence of good formal structures for flat meromorphic connections on surfaces after suitable blowing up; this verifies a conjecture of Sabbah and extends a result of Mochizuki for algebraic connections. Our proof uses a finiteness argument on the valuative tree associated to a point on a surface in order to verify the numerical criterion.

Layers of generality and types of generalization in pattern activities

Abstract: Pattern generalization is considered one of the prominent routes for introducing students to algebra. However, not all generalizations are algebraic. In the use of pattern generalization as a route to algebra, we �teachers and educators� thus have to remain vigilant in order not to confound algebraic generalizations with other forms of dealing with the general. But how to distinguish between algebraic and non-algebraic generalizations? On epistemological and semiotic grounds, in this article I suggest a characterization of algebraic generalizations. This characterization helps to bring about a typology of algebraic and arithmetic generalizations. The typology is illustrated with classroom examples.

Generalized q -Onsager Algebras and Boundary Affine Toda Field Theories

Abstract: Generalizations of the q-Onsager algebra are introduced and studied. In one of the simplest case and q = 1, the algebra reduces to the one proposed by Uglov–Ivanov. In the general case and q ≠ 1, an explicit algebra homomorphism associated with coideal subalgebras of quantum affine Lie algebras (simply and non-simply laced) is exhibited. Boundary (soliton non-preserving) integrable quantum Toda field theories are then considered in light of these results. For the first time, all defining relations for the underlying non-Abelian symmetry algebra are explicitly obtained. As a consequence, based on purely algebraic arguments all integrable (fixed or dynamical) boundary conditions are classified.

Automatic invariant generation for hybrid systems using ideal fixed points

Abstract: We present computational techniques for automatically generating algebraic (polynomial equality) invariants for algebraic hybrid systems. Such systems involve ordinary differential equations with multivariate polynomial right-hand sides. Our approach casts the problem of generating invariants for differential equations as the greatest fixed point of a monotone operator over the lattice of ideals in a polynomial ring. We provide an algorithm to compute this monotone operator using basic ideas from commutative algebraic geometry. However, the resulting iteration sequence does not always converge to a fixed point, since the lattice of ideals over a polynomial ring does not satisfy the descending chain condition.We then present a bounded-degree relaxation based on the concept of "pseudo ideals", due to Colon, that restricts ideal membership using multipliers with bounded degrees. We show that the monotone operator on bounded degree pseudo ideals is convergent and generates fixed points that can be used to generate useful algebraic invariants for non-linear systems. The technique for continuous systems is then extended to consider hybrid systems with multiple modes and discrete transitions between modes.We have implemented the exact, non-convergent iteration over ideals in combination with the bounded degree iteration over pseudo ideals to guarantee convergence. This has been applied to automatically infer useful and interesting polynomial invariants for some benchmark non-linear systems.

ABC: algebraic bound computation for loops

Abstract: We present ABC, a software tool for automatically computing symbolic upper bounds on the number of iterations of nested program loops. The system combines static analysis of programs with symbolic summation techniques to derive loop invariant relations between program variables. Iteration bounds are obtained from the inferred invariants, by replacing variables with bounds on their greatest values. We have successfully applied ABC to a large number of examples. The derived symbolic bounds express non-trivial polynomial relations over loop variables. We also report on results to automatically infer symbolic expressions over harmonic numbers as upper bounds on loop iteration counts.

Period domains over finite and p-adic fields

Abstract: This book is, on the one hand, a pedagogical introduction to the formalism of slopes, of semi-stability and of related concepts in the simplest possible context. It is therefore accessible to any graduate student with a basic knowledge in algebraic geometry and algebraic groups. On the other hand, the book also provides a thorough introduction to the basics of period domains, as they appear in the geometric approach to local Langlands correspondences and in the recent conjectural p-adic local Langlands program. The authors provide numerous worked examples and establish many connections to topics in the general area of algebraic groups over finite and local fields. In addition, the end of each section includes remarks on open questions, historical context and references to the literature.

Effective algebraic degeneracy

Abstract: We show that for every smooth projective hypersurface X⊂ℙn+1 of degree d and of arbitrary dimension n ≥2, if X is generic, then there exists a proper algebraic subvariety Y ⊊ X such that every nonconstant entire holomorphic curve f :ℂ→X has image f(ℂ) which lies in Y, as soon as its degree satisfies the effective lower bound $d\geqslant 2^{n^{5}}$ .

On Algebraic Expressions of Sigma Functions for (n, s) Curves

Abstract: An expression of the multivariate sigma function associated with an (n,s)-curve is given in terms of algebraic integrals. As a corollary the first term of the series expansion around the origin of the sigma function is directly proved to be Schur function determined from the gap sequence at infinity.

Algebraic pseudorandom functions with improved efficiency from the augmented cascade

Abstract: We construct an algebraic pseudorandom function (PRF) that is more efficient than the classic Naor-Reingold algebraic PRF. Our PRF is the result of adapting the cascade construction, which is the basis of HMAC, to the algebraic settings. To do so we define an augmented cascade and prove it secure when the underlying PRF satisfies a property called parallel security. We then use the augmented cascade to build new algebraic PRFs. The algebraic structure of our PRF leads to an efficient large-domain Verifiable Random Function (VRF) and a large-domain simulatable VRF.

Triangular Decomposition of Semi-algebraic Systems

Abstract: Regular chains and triangular decompositions are fundamental and well-developed tools for describing the complex solutions of polynomial systems. This paper proposes adaptations of these tools focusing on solutions of the real analogue: semi-algebraic systems. We show that any such system can be decomposed into finitely many {\em regular semi-algebraic systems}. We propose two specifications of such a decomposition and present corresponding algorithms. Under some assumptions, one type of decomposition can be computed in singly exponential time w.r.t.\ the number of variables. We implement our algorithms and the experimental results illustrate their effectiveness.

An algebraic proof of generalized Wick theorem

Abstract: The multireference normal order theory, introduced by Kutzelnigg and Mukherjee [J. Chem. Phys. 107, 432 (1997)], is defined explicitly, and an algebraic proof is given for the corresponding contraction rules for a product of any two normal ordered operators. The proof does not require that the contractions be cumulants, so it is less restricted. In addition, it follows from the proof that the normal order theory and corresponding contraction rules hold equally well if the contractions are only defined up to a certain level. These relaxations enable us to extend the original normal order theory. As a particular example, a quasi-normal-order theory is developed, in which only one-body contractions are present. These contractions are based on the one-particle reduced density matrix.

Newton Polyhedra of Discriminants of Projections

Abstract: For a system of polynomial equations, whose coefficients depend on parameters, the Newton polyhedron of its discriminant is computed in terms of the Newton polyhedra of the coefficients. This leads to an explicit formula (involving Euler obstructions of toric varieties) in the unmixed case, suggests certain open questions in general, and generalizes a number of similar known results (Gelfand et al. in Discriminants, resultants, and multidimensional determinants. Birkhauser, Boston, 1994; Sturmfels in J. Algebraic Comb. 32(2):207---236, 1994; McDonald in Discrete Comput. Geom. 27:501---529, 2002; Gonzalez-Perez in Can. J. Math. 52(2):348-368, 2000; Esterov and Khovanskii in Funct. Anal. Math. 2(1), 2008).

Tight Bounds for Algebraic Gossip on Graphs

Abstract: We study the stopping times of gossip algorithms for network coding. We analyze algebraic gossip (i.e., random linear coding) and consider three gossip algorithms for information spreading Pull, Push, and Exchange. The stopping time of algebraic gossip is known to be linear for the complete graph, but the question of determining a tight upper bound or lower bounds for general graphs is still open. We take a major step in solving this question, and prove that algebraic gossip on any graph of size n is O(Δn) where Δ is the maximum degree of the graph. This leads to a tight bound of Θ(n) for bounded degree graphs and an upper bound of O(n2) for general graphs. We show that the latter bound is tight by providing an example of a graph with a stopping time of Ω(n2). Our proofs use a novel method that relies on Jackson's queuing theorem to analyze the stopping time of network coding; this technique is likely to become useful for future research.

Rational first integrals in the Darboux theory of integrability in C^n

Abstract: In 1979 Jouanolou showed that if the number of invariant algebraic hypersurfaces of a polynomial vector field in R n or C n of degree d is at least ( d + n − 1 n ) + n , then the vector field has a rational first integral. His proof used sophisticated tools of algebraic geometry. We provide an easy and elementary proof of Jouanolou's result using linear algebra.