The algebraic degree of geometric optimization problems

In this paper we apply Galois methods to certain fundamentalgeometric optimization problems whose exact computational complexity has been an open problem for a long time. In particular we show that the classic Weber problem, along with theline-restricted Weber problem and itsthree-dimensional version are in general not solvable by radicals over the field of rationals. One direct consequence of these results is that for these geometric optimization problems there existsno exact algorithm under models of computation where the root of an algebraic equation is obtained using arithmetic operations and the extraction ofkth roots. This leaves only numerical or symbolic approximations to the solutions, where the complexity of the approximations is shown to be primarily a function of the algebraic degree of the optimum solution point.     

Saddle singularities of complexity 1 of integrable Hamiltonian systems

Properties of saddle singularities of rank 0 and complexity 1 for integrable Hamiltonian systems are studied. An invariant (f _n -graph) solving the problem of semi-local classification of saddle singularities of rank 0 for an arbitrary complexity was constructed earlier by the author. In this paper, a more simple form of the invariant for singularities of complexity 1 is obtained and some properties of such singularities are described in algebraic terms. In addition, the paper contains a list of saddle singularities of complexity 1 for systems with three degrees of freedom.     

Improving algebraic attacks on stream ciphers based on linear feedback shift register over $$\mathbb {F}_{2^k}$$

In this paper we investigate univariate algebraic attacks on filter generators over extension fields \(\mathbb {F}_q=\mathbb {F}_{2^n}\) with focus on the Welch–Gong (WG) family of stream ciphers. Our main contribution is to reduce the general algebraic attack complexity on such cipher by proving new and lower bounds for the spectral immunity of such ciphers. The spectral immunity is the univariate analog of algebraic immunity and instead of measuring degree of multiples of a multivariate polynomial, it measures the minimum number of nonzero coefficients of a multiple of a univariate polynomial. In particular, there is an algebraic degeneracy in these constructions, which, when combined with attacks based on low-weight multiples over \(\mathbb {F}_q\), provides much more efficient attacks than over \(\mathbb {F}_2\). With negligible computational complexity, our best attack breaks the primitive WG-5 if given access to 4 kilobytes of keystream, break WG-7 if given access to 16 kilobytes of keystream and break WG-8 if given access to half a megabyte of keystream. Our best attack on WG-16 targeted at 4G-LTE is less practical, and requires \(2^{103}\) computational complexity and \(2^{61}\) bits of keystream. In all instances, we significantly lower both keystream and computational complexity in comparison to previous estimates. On a side note, we resolve an open problem regarding the rank of a type of equation systems used in algebraic attacks.     

Computing arbitrary Lagrangian Eulerian maps for evolving surfaces

The good mesh quality of a discretized closed evolving surface is often compromised during time evolution. In recent years this phenomenon has been theoretically addressed in a few ways, one of them uses arbitrary Lagrangian Eulerian (ALE) maps. However, the numerical computation of such maps still remained an unsolved problem in the literature. An approach, using differential algebraic problems, is proposed here to numerically compute an arbitrary Lagrangian Eulerian map, which preserves the mesh properties over time. The ALE velocity is obtained by finding an equilibrium of a simple spring system, based on the connectivity of the nodes in the mesh. We also consider the algorithmic question of constructing acute surface meshes. We present various numerical experiments illustrating the good properties of the obtained meshes and the low computational cost of the proposed approach.     

A polynomial-time complexity bound for the computation of the singular part of a Puiseux expansion of an algebraic function

In this paper we present a refined version of the Newton polygon process to compute the Puiseux expansions of an algebraic function defined over the rational function field. We determine an upper bound for the bit-complexity of computing the singular part of a Puiseux expansion by this algorithm, and use a recent quantitative version of Eisenstein's theorem on power series expansions of algebraic functions to show that this computational complexity is polynomial in the degrees and the logarithm of the height of the polynomial defining the algebraic function.

     

Curves with Many Points and Multiplication Complexity in Any Extension of q

From the existence of algebraic function fields having some good properties, we obtain some new upper bounds on the bilinear complexity of multiplication in all extensions of the finite field _q, where q is an arbitrary prime power. So we prove that the bilinear complexity of multiplication in the finite fields _qⁿ is linear uniformly in q with respect to the degree n.     

Computability of String Functions Over Algebraic Structures Armin Hemmerling

We present a model of computation for string functions over single-sorted, total algebraic structures and study some basic features of a general theory of computability within this framework. Our concept generalizes the Blum-Shub-Smale setting of computability over the reals and other rings. By dealing with strings of arbitrary length instead of tuples of fixed length, some suppositions of deeper results within former approaches to generalized recursion theory become superfluous. Moreover, this gives the basis for introducing computational complexity in a BSS-like manner. Relationships both to classical computability and to Friedman's concept of eds computability are established. Two kinds of nondeterminism as well as several variants of recognizability are investigated with respect to interdependencies on each other and on properties of the underlying structures. For structures of finite signatures, there are universal programs with the usual characteristics. For the general case of not necessarily finite signature, this subject will be studied in a separate, forthcoming paper.     

Polynomial complexity algorithms for computational problems in the theory of algebraic curves

It is proved that the classical algorithm for constructing Newton-Puiseux expansions for the roots of polynomials using the method of Newton polygons is of polynomial complexity in the notation length of the expansion coefficients. This result is used, in the case of a ground field of characteristic O, to construct polynomial-time algorithms for factoring polynomials over fields of formal power series, and for fundamental computational problems in the theory of algebraic curves, such as curve normalization.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 176, pp. 127–150, 1989.     

Die Berechnungskomplexität von elementarsymmetrischen Funktionen und von Interpolationskoeffizienten

Summary n(lg₂ n–2) multiplications and divisions are necessary to compute the set of elementary symmetric functions inn indeterminates. This lower bound and similar ones for the computational complexity of various evaluation and interpolation problems are obtained by introducing ideas and results from algebraic geometry.     

Complexity and rank of homogeneous spaces

We study an algebraic varieties with the action of a reductive group G. The relation is elucidated between the notions of complexity and rank of an arbitrary G-variety and the structure of stabilizers of general position of some actions of G itself and its Borel subgroup. The application of this theory to homogeneous spaces provides the explicit formulas for the rank and the complexity of quasiaffine G/H in terms of co-isotropy representation of H. The existence of Cartan subspace (and hence the freeness of algebra of invariants) for co-isotropy representation of a connected observable spherical subgroup H is proved.     

OBSERVATIONS ON INFEASIBILITY DETECTORS FOR CLASSIFYING CONIC CONVEX PROGRAMS

Abstract

New insights are presented from a convex duality classification theory of over 25 years ago for recent computational complexity results developed for recognizing some of the asymptotic duality states that prevail for an arbitrary convex programming pair.     

Lower bounds for algebraic complexity of nilpotent associative algebras

Exact algebraic algorithms for calculating the product of two elements of nilpotent associative algebras over fields of characteristic zero are considered (this is a particular case of simultaneous calculation of several multinomials). The complexity of an algebra in this computational model is defined as the number of nonscalar multiplications of an optimal algorithm. Lower bounds for the tensor rank of nilpotent associative algebras (in terms of dimensions of certain subalgebras) are obtained, which give lower bounds for the algebraic complexity of this class of algebras. Examples of reaching these estimates for different dimensions of nilpotent algebras are presented.     

Short Proofs Using Compact Representations of Algebraic Integers

We show that under the assumption of a certain generalized Riemann hypothesis the problem of verifying the value of the class number of an arbitrary algebraic number field of arbitrary degree belongs to the complexity class NP. In order to prove this result we introduce compact representations of algebraic integers which allows us to represent a system of fundamental units by (log₂(Δ))^O(1) bits.     

The castelnuovo criterion of rationality

A new proof is provided for the rationality criterion for algebraic surfaces over an arbitrary base field, usingl-adic cohomologies.Translated from Matematicheskie Zametki, Vol.11, No. 1, pp. 27–32, January, 1972.     

Isotropic Trialitarian Algebraic Groups

By modifying a construction from Knuset al., we construct all isotropic algebraic groups of type³D₄and⁶D₄over an arbitrary field of characteristic ≠ 2. We also provide a nice isomorphism criterion for such groups. The results of this paper extend the main results of Allison (using entirely different methods) to fields of nonzero characteristic and algebraic groups.     

Poles of Intertwining Operators via Endoscopy; the Connection with Prehomogeneous Vector Spaces With an Appendix, `Basic Endoscopic Data', by Diana Shelstad To the Memory of Magdy Assem

In this paper, we determine the residues at poles of standard intertwining operators for parabolically induced representations of an arbitrary connected reductive quansisplit algebraic group over a p-acid field whenever the unipotent radical of the parabolic subgroup is Abelian. We then interpret these residues by means of the theory of endoscopy.     

The Hardness of Polynomial Equation Solving

Elimination theory was at the origin of algebraic geometry in the nineteenth century and now deals with the algorithmic solving of multivariate polynomial equation systems over the complex numbers or, more generally, over an arbitrary algebraically closed field. In this paper we investigate the intrinsic sequential time complexity of universal elimination procedures for arbitrary continuous data structures encoding input and output objects of elimination theory (i.e., polynomial equation systems) and admitting the representation of certain limit objects. Our main result is the following: let there be given such a data structure and together with this data structure a universal elimination algorithm, say P, solving arbitrary parametric polynomial equation systems. Suppose that the algorithm P avoids unnecessary branchings and that P admits the efficient computation of certain natural limit objects (as, e.g., the Zariski closure of a given constructible algebraic set or the parametric greatest common divisor of two given algebraic families of univariate polynomials). Then P$ cannot be a polynomial time algorithm. The paper contains different variants of this result and discusses their practical implications.     

Affine algebraic sets relative to an algebraic theory

For an arbitrary algebraic theory T and a prescribed T-algebraL, the geometrical category AfAlgSet(L) of affine algebraic sets over the affine lineL is build up. It is proved to be dually equivalent to the category FcAlg(L) of functional T-algebras overL. The Galois theory ofL is defined and a Galois duality established. These geometrical categories are characterized up to an equivalence of categories.     

Low increasing tower of algebraic function fields and bilinear complexity of multiplication in any extension of

From the existence of a tower of algebraic function fields with more steps than the Garcia–Stichtenoth tower, we improve upper bounds on the bilinear complexity of multiplication in all extensions of the finite field where q is an arbitrary prime power.