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Algebraic tori occupy a special place among linear algebraic groups. An algebraic torus can be defined over an arbitrary field but if the ground field is of arithmetic type, one can additionally consider schemes over the ring of integers of this field, which are related to the original tori and called their integral models. The Neron and Voskresenskiĭ models are most well known among them. There exists a broad range of problems dealing with the construction of these models and the elucidation of their properties. This paper is devoted to the study of the main integral models of algebraic tori over fields of algebraic numbers, to the comparison of their properties, and to the clarification of links between them. At the end of this paper, a special family of maximal algebraic tori unaffected inside semisimple groups of Bn type is presented as an example for realization of previously investigated constructions.

Let $X$ be a connected affine homogenous space of a linear algebraic group $G$ over $\C$. (1) If $X$ is different from a line or a torus we show that the space of all algebraic vector fields on $X$ coincides with the Lie algebra generated by complete algebraic vector fields on $X$. (2) Suppose that $X$ has a $G$-invariant volume form $\omega$. We prove that the space of all divergence-free (with respect to $\omega$) algebraic vector fields on $X$ coincides with the Lie algebra generated by divergence-free complete algebraic vector fields on $X$ (including the cases when $X$ is a line or a torus). The proof of these results requires new criteria for algebraic (volume) density property based on so called module generating pairs., Comment: 21 pages

Let K be a number field of degree n over ℚ and let d, h, and R be the absolute values of the discriminant, class number, and regulator of K, respectively. It is known that if K contains no quadratic subfield, then $$ h\;R\gg \frac{d^{1/2}}{ \log d}, $$ where the implied constant depends only on n. In Theorem 1, this lower estimate is improved for pure cubic fields. Consider the family $$ {\mathcal{K}}_n $$ , where K ∈ $$ {\mathcal{K}}_n $$ if K is a totally real number field of degree n whose normal closure has the symmetric group S n as its Galois group. In Theorem 2, it is proved that for a fixed n ≥ 2, there are infinitely many K ∈ $$ {\mathcal{K}}_n $$ with $$ h\gg {d}^{1/2}{\left( \log \log d\right)}^{n-1}/{\left( \log d\right)}^n, $$ where the implied constant depends only on n. This somewhat improves the analogous result h ≫ d 1/2/(log d) n of W. Duke [MR 1966783 (2004g:11103)].

A power moments based algebraic method that takes into account the local Taylor’s expansion structure of the wave function is applied to find the spectrum for the two dimensional parity-time symmetric potential V(x, y) = x 2 + y 2 + i g x 2 y. Converging results are presented for a wide range of the strength parameter g.

In this paper, two constructions of Boolean functions with optimal algebraic immunity are proposed. They generalize previous ones respectively given by Rizomiliotis (IEEE Trans Inf Theory 56:4014---4024, 2010) and Zeng et al. (IEEE Trans Inf Theory 57:6310---6320, 2011) and some new functions with desired properties are obtained. The functions constructed in this paper can be balanced and have optimal algebraic degree. Further, a new lower bound on the nonlinearity of the proposed functions is established, and as a special case, it gives a new lower bound on the nonlinearity of the Carlet-Feng functions, which is slightly better than the best previously known ones. For $$n\le 19$$n≤19, the numerical results reveal that among the constructed functions in this paper, there always exist some functions with nonlinearity higher than or equal to that of the Carlet-Feng functions. These functions are also checked to have good behavior against fast algebraic attacks at least for small numbers of input variables.

Let G be a commutative algebraic group over a finitely generated infinite field K of characteristic p. We prove that every extension of K contained in the field obtained by adjoining to K all prime-to-p torsion points of G is Hilbertian. We also determine when the field obtained by adjoining to K all torsion points of G has this property. This extends results of Moshe Jarden on abelian varieties.

If G is a connected linear algebraic group over the field k, a Levi factor of G is a reductive complement to the unipotent radical of G. If k has positive characteristic, G may have no Levi factor, or G may have Levi factors which are not geometrically conjugate. We give in this paper some sufficient conditions for the existence and the conjugacy of Levi factors of G. Let A be a Henselian discrete valuation ring with fractions K and with perfect residue field k of characteristic p>0. Let G be a connected and reductive algebraic group over K. Bruhat and Tits have associated to G certain smooth A-group schemes P whose generic fibers P/K coincide with G; these are known as *parahoric group schemes*. The special fiber P/k of a parahoric group scheme is a linear algebraic group over k. If G splits over an unramified extension of K, we show that P/k has a Levi factor, and that any two Levi factors of P/k are geometrically conjugate., To appear, Morozov issue of Transformation Groups

The sixteen families of q-series containing the Ramanujan functions were listed by I.J. Zucker (SIAM J. Math. Anal. 10:192–206, 1979), which are generated from the Fourier series expansions of the Jacobian elliptic functions or some of their squares. This paper discusses algebraic independence properties for these q-series. We determine all the sets of q-series such that, at each algebraic point, the values of the q-series in the set are algebraically independent over ℚ. We also present several algebraic relations over ℚ for two or three of these q-series.

Algebraic immunity (AI) measures the resistance of a Boolean function f against algebraic attack. Extended algebraic immunity (EAI) extends the concept of algebraic immunity, whose point is that a Boolean function f may be replaced by another Boolean function f c called the algebraic complement of f. In this paper, we study the relation between different properties (such as weight, nonlinearity, etc.) of Boolean function f and its algebraic complement f c . For example, the relation between annihilator sets of f and f c provides a faster way to find their annihilators than previous report. Next, we present a necessary condition for Boolean functions to be of the maximum possible extended algebraic immunity. We also analyze some Boolean functions with maximum possible algebraic immunity constructed by known existing construction methods for their extended algebraic immunity.

In the past few years, algebraic attacks against stream ciphers with linear feedback function have been significantly improved. As a response to the new attacks, the notion of algebraic immunity of a Boolean function f was introduced, defined as the minimum degree of the annihilators of f and f + 1. An annihilator of f is a nonzero Boolean function g, such that fg = 0. There is an increasing interest in construction of Boolean functions that possess optimal algebraic immunity, combined with other characteristics, like balancedness, high nonlinearity, and high algebraic degree. In this paper, we investigate a recently proposed infinite class of balanced Boolean functions with optimal algebraic immunity, optimum algebraic degree and much better nonlinearity than all the previously introduced classes of Boolean functions with maximal algebraic immunity. More precisely, we study the resistance of the functions against one of the new algebraic attacks, namely the fast algebraic attacks (FAAs). Using the special characteristics of the family members, we introduce an efficient method for the evaluation of their behavior against these attacks. The new algorithm is based on the well studied Berlekamp---Massey algorithm.

Let G be an affine algebraic group and let X be an affine algebraic variety. An action G × X → X is called observable if for any G-invariant, proper, closed subset Y of X there is a nonzero invariant f ∈ \(\Bbbk\) [X]G such that f|Y = 0. We characterize this condition geometrically as follows. The action G × X → X is observable if and only if

In [4] we have introduced a new distance between Galois orbits over ℚ. Using generalized divisors, we have extended the notion of trace of an algebraic number to other transcendental quantities. In this article we continue the work started in [4]. We define the critical function for a class of transcendental numbers, that generalizes the notion of minimal polynomial of an algebraic number. Our results extend the results obtained by Popescu et al [5].

We study distribution of irreducible algebraic integers in short intervals and prove that if the class number of an algebraic number field K exceeds 2, every interval of the form (x, x + x θ) with a fixed θ > 1/2 contains absolute value of the norm of an irreducible algebraic integer from K provided x ≥ x 0. The constant x 0 effectively depends on K and θ.

Algebraic immunity is a new cryptographic criterion proposed against algebraic attacks. In order to resist algebraic attacks, Boolean functions used in many stream ciphers should possess high algebraic immunity. This paper presents two main results to find balanced Boolean functions with maximum algebraic immunity. Through swapping the values of two bits, and then generalizing the result to swap some pairs of bits of the symmetric Boolean function constructed by Dalai, a new class of Boolean functions with maximum algebraic immunity are constructed. Enumeration of such functions is also given. For a given function p(x) with deg(p(x)) < Open image in new window, we give a method to construct functions in the form p(x)+q(x) which achieve the maximum algebraic immunity, where every term with nonzero coefficient in the ANF of q(x) has degree no less than Open image in new window.

Let G be a p-adic algebraic group of polynomial growth and H be a closed subgroup of G. We prove the growth conjecture for the homogeneous space G/H, that is, G/H supports a recurrent random walk if and only if G/H has polynomial growth of degree atmost two.

Suppose that D is a division ring with center F and N is a non-central normal subgroup of GLn(D). In this paper we generalize some known results about maximal subgroups of GLn(D) to maximal subgroups of N. More precisely, we prove that if M is a maximal subgroup of N such that F[M] satisfies a polynomial identity and \(C_{M_{n}(D)}(M) \backslash F\) contains an algebraic element over F or \(C_{M_{n}(D)}(M)=F\) and either n ≥ 2 or n = 1 and M is not abelian, then [D : F] < ∞.

Let k be an algebraically closed base field of arbitrary characteristic. In this paper, we study actions of a connected solvable linear algebraic group G on a central simple algebra Q. The main result is the following: Q can be split G-equivariantly by a finite-dimensional splitting field, provided that G acts “algebraically,” i.e., provided that Q contains a G-stable order on which the action is rational. As an application, it is shown that rational torus actions on prime PI-algebras are induced by actions on commutative domains.

This paper shows that for a local field K, a subfield k ⊂ K and a variety X over k, X is complete if and only if for every finite field extension Kʹ | K, the set X(Kʹ) is compact in its strong topology.

We compare three levels of algebraic certificates for evaluating the maximum mod- ulus of a complex analytic polynomial, on a compact semi-algebraic set. They are obtained as translations of some recently discovered inequalities in operator theory. Although they can be stated in purely algebraic terms, the only known proofs for these decompositions have a transcen- dental character.

We examine families of twists by an automorphism of the complex polynomial ring on n generators. The multiplication in the twisted algebra determines a Poisson structure on affine n-space. We demonstrate that if the automorphism has a single eigenvalue, then the primitive ideals in the twist are parameterized by the algebraic symplectic leaves associated to this Poisson structure. Furthermore, in this case all of the leaves are algebraic and can be realized as the orbits of an algebraic group.

Consider real polynomials g 1, . . . , g r in n variables, and assume that the subset K = {g 1≥0, . . . , g r ≥0} of ℝ n is compact. We show that a polynomial f has a representation in which the s e are sums of squares, if and only if the same is true in every localization of the polynomial ring by a maximal ideal. We apply this result to provide large and concrete families of cases in which dim (K) = 2 and every polynomial f with f| K ≥0 has a representation (*). Before, it was not known whether a single such example exists. Further geometric and arithmetic applications are given.

The algebraic independence of $$ e^{{\theta _{1} }} , \ldots ,e^{{\theta _{s} }} $$ is proved, where θ1, . . . , θs are certain gap series or power series of algebraic numbers, or certain transcendental continued fractions with algebraic elements.

Let T be an order bounded disjointness preserving operator on an Archimedean vector lattice. The main result in this paper shows that T is algebraic if and only if there exist natural numbers m and n such that n ≥ m, and Tn!, when restricted to the vector sublattice generated by the range of Tm, is an algebraic orthomorphism. Moreover, n (respectively, m) can be chosen as the degree (respectively, the multiplicity of 0 as a root) of the minimal polynomial of T. In the process of proving this result, we define strongly diagonal operators and study algebraic order bounded disjointness preserving operators and locally algebraic orthomorphisms. In addition, we introduce a type of completeness on Archimedean vector lattices that is necessary and sufficient for locally algebraic orthomorphisms to coincide with algebraic orthomorphisms.

Bounded algebraic sets over a free Lie algebra F over a field k are classified in three equivalent languages: (1) in terms of algebraic sets; (2) in terms of radicals of algebraic sets; (3) in terms of coordinate algebras of algebraic sets.

In the present paper, we consider products of lengths of algebraic numbers whose sum or product is a chosen algebraic number. These products are used to construct a new height function for algebraic numbers. With the help of this function, a metric on the set of all algebraic numbers, which induces the discrete topology, is introduced.

In Geometriae Dedicata 79 (2000), 101–108, Rudolf Winkel conjectured: for a given algebraic curve f=0 of degree m ≥ 4 there is in general no polynomial vector field of degree less than 2m -1 leaving invariant f=0 and having exactly the ovals of f=0 as limit cycles. Here we show that this conjecture is not true.

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 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., Comment: 82 pages, submitted to Foundations of Computational Mathematics

Let K be a fixed totally real algebraic number field of finite degree over the rationals. The theme of this paper is the problem about the occurrence of algebraic almost-primes in a polynomial sequence generated by an irreducible polynomial of K with prime arguments. The method is based on a weighted upper and lower linear Selberg-type sieve in K and makes use of a multidimensional algebraic version of Bombieri’s theorem on primes in arithmetic progressions.

Given a finite field K, we prove that every algebraic number over K can be expressed by both linear and multiplicative forms in conjugates over K of another algebraic number. Furthermore, it is shown that every linear form over K in conjugates vanishes for some nonzero algebraic number. A multiplicative analogue of this result is also obtained.

In this paper, by using the method of algebraic analysis, the results in our previous work are generalized. These results are of importance in the qualitative theory of polynomial autonomous systems.

We study the computational complexity of linear programs with coefficients that are real algebraic numbers under a Turing machine model of computation. After reviewing a method for exact representation of algebraic numbers under the Turing model, we show that the fundamental tasks of comparison and arithmetic can be performed in polynomial time. Our technique for establishing polynomial-time algorithms for comparison and arithmetic is distinct from the usual resultant-based approaches, and has the advantage that it provides a natural framework for analysis of the complexity of computational tasks, such as Gaussian elimination, that involve a sequence of arithmetic operations. Our main contribution is to show that a variant of the ellipsoid method can be used to solve linear programming in time polynomial in the encoding size of the problem coefficients and the degree of any algebraic extension that contains those coefficients.

— We prove an algebraicity criterion for leaves of algebraic foliations defined over number fields. Namely, consider a number field K embedded in C, a smooth algebraic variety X over K, equipped with a K-rational point P, and F an algebraic subbundle of the its tangent bundle TX, defined over K. Assume moreover that the vector bundle F is involutive, i.e., closed unter Lie bracket. Then it defines an holomorphic foliation of the analytic mainfold X(C), and one may consider its leaf ℱ through P. We prove that ℱ is algebraic if the following local conditions are satisfied: i) For almost every prime ideal p of the ring of integers 𝒪K of the number field K, the p-curvature of the reduction modulo p of the involutive bundle F vanishes at P (where p denotes the characteristic of the residue field 𝒪K / p ). ii) The analytic manifold ℱ satisfies the Liouville property; this arises, in particular, if ℱ is the image by some holomorphic map of the complement in a complex algebraic variety of a closed analytic subset. This algebraicity criterion unifies and extends various results of D. V. and G. V. Chudnovsky, Andre, and Graftieaux, and also admits new consequences. For instance, applied to an algebraic group G over K, it shows that a K-Lie subalgebra h of Lie G is algebraic if and only if for almost every non-zero prime ideal p of 𝒪K , of residue characteristic p, the reduction modulo p of h is a restricted Lie subalgebra of the reduction modulo p of Lie G (i.e., is stable under p-th powers). This solves a conjecture of Ekedahl and Shepherd-Barron. The algebraicity criterion above follows from a more basic algebraicity criterion concerning smooth formal germs in algebraic varieties over number fields. The proof of the latter relies on “transcendence techniques”, recast in a modern geometric version involving elementary concepts of Arakelov geometry, and on some analytic estimates, related to the First Main Theorem of higher-dimensional Nevanlinna theory.

For each non-singular real algebraic curve f = 0 of degree m we exhibit an explicit vector field of degree m which has precisely the bounded components of f = 0 as limit cycles. The degree of the system is optimal for a generic class of algebraic curves and improves the significantly the bounds given by Winkel.

Let k(X)/k be a finitely generated purely transcendental extension of fields with transcendence basis X. From a generalization of Luroth's theorem it is known that all intermediate fields K of k(X)/k with (K/k) = 1 are of the form K = k(f) for some f∈k(X). In this note we give a criterion which allows to decide effectively whether the extension K/k is simple if a finite generating set of K over k is known and computations in K are effective. In the affirmative case a primitive element of K over k is computed.

By the use of approximation method a general criterion of algebraic independence of complex numbers is established. As its application the algebraic independence of values of certain gap series at algebraic and transcendental points is given.

In this note the algebraic independence of values of general Mahler series with different parameters at algebraic points is established.

For a collection ofE-functions which is algebraically dependent over the field of rational functions, theorems on the algebraic independence of values of subcollections at algebraic points are proved.

We study the question of which torsion subgroups of commutative algebraic groups over finite fields are contained in modular difference algebraic groups for some choice of a field automorphism. We show that if G is a simple commutative algebraic group over a finite field of characteristic p, ? is a prime different from p, and for some difference closed field (?, σ) the ?-primary torsion of G(?) is contained in a modular group definable in (?, σ), then it is contained in a group of the form {x∈G(?) :σ(x) =[a](x) } with a∈ℕ\pℕ. We show that no such modular group can be found for many G of interest. In the cases that such equations may be found, we recover an effective version of a theorem of Boxall.

We present an introductory survey to the Darboux integrability theory of planar complex and real polynomial differential systems. Our presentation contains some improvements to the classical theory.

In this note we will prove an algebraic characterization of the piecewiese polynomial functions of a real ℚ-algebra A. This characterization is related to the realm of investigations concerning the Pierce–Birkhoff conjecture.

Effective rational and algebraic approximations of a large class of algebraic numbers are obtained by Thue-Siegel’s method. As an application of this result, it is proved that: if D>0 is not a square, and e =x0 denotes the fundamental solution ofx2−Dy2=−1, thenx2+1=Dy4 is solvable if and only ify0=A2, where A is an integer. Moreover, if e>64, thenx2+1=Dy4 has at most one positive integral solution (x, y).

The Gorbunov-Tumanov conjecture on the structure of lattices of quasivarieties is proved true for the case of algebraic lattices. Namely, for an algebraic atomistic lattice L, the following conditions are equivalent: (1) L is represented as Lq(K) for some algebraic quasivariety K; (2) L is represented as SΛ (A) for some algebraic lattice A which satisfies the minimality condition and nearly satisfies the maximality conditions; (3) L is a coalgebraic lattice admitting an equaclosure operator.

Let v be a valuation of a field K with a value group and a residue field , w be an extension of v to . Then w is called a residual algebraic torsion extension of v to if is an algebraic extension and is a torsion group. In this paper, a residual algebraic torsion extension of v to is described and its certain properties are investigated. Also, the existence of a residual algebraic torsion extension of a valuation on K to with given residue field and value group is studied. MSC:12J10, 12J20, 12F20.

Algebraic properties of functional matrices arising in the constuction of graded Pade approximations are established. This construction plays an important role in the theory of transcendental numbers.

In this paper we show that for a formally real fieldK such that 3 is a square inK and such that P2 (K)=2 the inequality P4 (K)≤6 holds.