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The relation between soft element-wise field and soft int-field has been established and then some properties of soft int-field are studied. We define the notions of soft algebraic element and soft purely inseparable element of a soft int-field extension. Some characterizations of soft algebraic and soft purely inseparable int-field extensions are given. Lastly, we define soft separable algebraic int-field extension and study some of its properties.

In this paper, we first prove that given pairwise distinct algebraic numbers $\alpha_1, \ldots, \alpha_n$, the numbers $\alpha_1+t, \ldots, \alpha_n+t$ are multiplicatively independent for all sufficiently large integers $t$. Then, for a pair $(a,b)$ of distinct integers, we study how many pairs $(a+t,b+t)$ are multiplicatively dependent when $t$ runs through the integers. For such a pair $(a,b)$ with $b-a=30$ we show that there are $13$ integers $t$ for which the pair $(a+t,b+t)$ is multiplicatively dependent. We conjecture that $13$ is the largest value of such translations for any $(a,b)$, where $a \ne b$, prove this for all pairs $(a,b)$ with difference at most $10^{10}$, and, assuming that the $ABC$ conjecture is true, show that for any such pair $(a,b)$, $a \ne b$, there is an absolute bound $C_1$ (independent of $a$ and $b$) on the number of such translations $t$., Comment: 22 pages

Algebraic independence of values of certain infinite products is proved, where the transcendence of such numbers was already established by Tachiya. As applications explicit examples of algebraically independent numbers are also given.

Let K be an algebraic extension of a field k, let σ = (σij) be an irreducible full (elementary) net of order n ≥ 2 (respectively, n ≥ 3) over K, while the additive subgroups σij are k-subspaces of K. We prove that all σij coincide with an intermediate subfield P, k ⊆ P ⊆ K, up to conjugation by a diagonal matrix.

Many mathematicians have studied the algebraic independence over $\mathbb{Q}$ of the values of gap series, and the values of lacunary series satisfying functional equations of Mahler type. In this paper, we give a new criterion for the algebraic independence over $\mathbb{Q}$ of the values $\sum^{\infty}_{n=0} t(n) \beta^{-n}$ for distinct sequences $(t(n))^{\infty}_{n=0}$ of nonnegative integers, where $\beta$ is a fixed Pisot or Salem number. Our criterion is applicable to certain power series which are not lacunary. Moreover, our criterion does not use functional equations. Consequently, we deduce the algebraic independence of certain values $\sum^{\infty}_{n=0} t_1 (n) \beta^{-n} , \dotsc , \sum^{\infty}_{n=0} t_r( n) \beta^{-n}$ satisfying $$\lim_{n \to \infty , t{i-1} (n) \neq 0} \; \dfrac{t_i(n)}{t_{i-1}(n)^M} = \infty \; (i=2, \dotsc, r)$$ for any positive real number $M$.

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 AK denote the ring of algebraic integers of an algebraic number field K=Q(θ) where the algebraic integer θ has minimal polynomial F(x)=xn+axm+b over the field Q of rational numbers with n=mt+u, t∈N, 0≤u≤m−1. In this paper, we characterize those primes which divide the discriminant of F(x) but do not divide [AK:Z[θ]] when u=0 or u divides m; such primes p are important for explicitly determining the decomposition of pAK into a product of prime ideals of AK in view of the well known Dedekind theorem. As a consequence, we obtain some necessary and sufficient conditions involving only a, b, m, n for AK to be equal to Z[θ].

Let P k : = F 2 [ x 1 , x 2 , … , x k ] be the graded polynomial algebra over the prime field of two elements F 2 , in k generators x 1 , x 2 , … , x k , each of degree 1. Being the mod-2 cohomology of the classifying space B ( Z / 2 ) k , the algebra P k is a module over the mod-2 Steenrod algebra A . In this Note, we extend a result of Hưng on Kameko's homomorphism S q ˜ ⁎ 0 : F 2 ⊗ A P k ⟶ F 2 ⊗ A P k . Using this result, we show that Singer's conjecture for the algebraic transfer is true in the case k = 5 and the degree 7.2 s − 5 with s an arbitrary positive integer.

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

The main purpose of this paper is to study the problems on the existence of algebraic solutions for some second-order complex differential equations with entire algebraic function element coefficients. Several theorems on the existence of solutions are obtained, which perfect the solution theory of linear complex differential equations.

In this note, we consider the special values of [Formula: see text]-analogues of Dirichlet [Formula: see text]-functions, namely, the values of the functions [Formula: see text] at positive integers [Formula: see text], where [Formula: see text] is a primitive Dirichlet character and [Formula: see text] is a complex number such that [Formula: see text]. We prove that if [Formula: see text] and [Formula: see text] is algebraic, then [Formula: see text] is transcendental. We also prove that if [Formula: see text] and [Formula: see text] is algebraic, then there exists a transcendental number [Formula: see text] which depends only on [Formula: see text] and is [Formula: see text]-linearly independent with [Formula: see text] such that [Formula: see text] is algebraic. These results can be viewed as an analogue of the classical result of Hecke on the arithmetic nature of the special values [Formula: see text] for [Formula: see text].

Altinel and Cherlin proved that any perfect central extension of a simple algebraic group over an algebraically closed field which happens to be of finite Morley rank is actually a finite central extension and is itself an algebraic group. We will prove an infinite rank version of their result with an additional hypothesis, while giving an example which shows the necessity of this hypothesis. The inspiration for the work comes from differential algebra; namely, a differential algebraic version of the results here was used by the second author to answer a question of Cassidy and Singer. The work here also provides an alternate path to the same answer.An almost simple superstable group G is one in which every definable normal subgroup H has the property that RU ( G ) RU ( H ) ? n for any n ? N . We prove that any almost simple superstable group which is a central extension of a simple algebraic group is actually a finite central extension and is an algebraic group. We also explain the applications of this result to differential algebraic groups. Many of the central ideas of the proof of our main theorem are an adaptation of techniques developed by the second author in the setting of differential algebraic groups.

In this study the residual algebraic free extension of a valuation on a field K to K(x) is studied. It is assumed that v is a valuation on K with rankv = 2 and the residual algebraic free extension w of v to K(x) with rankw = 3 is defined for a special case.

In this paper, we study about trigonometry in finite field, we know that , the field with p elements, where p is a prime number if and only if p = 8k + 1 or p = 8k -1. Let F and K be two fields, we say that F is an extension of K, if K⊆F or there exists a monomorphism f: K→F. Recall that , F[x] is the ring of polynomial over F. If (means that F is an extension of K), an element is algebraic over K if there exists such that f(u) = 0 (see [1]-[4]). The algebraic closure of K in F is , which is the set of all algebraic elements in F over K.

Unless otherswise specified, all objects are defined over a field k of characteristic 0. Let K be a field. The unessentialness of an extension of the algebra Der K by means of a splittable semisimple Lie algebra is established. Let D K be the category of differential Lie algebras (DL-algebras) (g;K). In this paper for an extension L/K the functor η:D K → D L , defining the tensor product L ⊗ K of vector spaces and the homomorphism of Lie algebras, is constructed. If the extension L/K is algebraic, then η is unique. The results will be required for strengthening the progress on Gelfand–Kirillov problem and weakened conjecture [1, 2].

In this paper we characterize all cubic polynomial differential systems in the plane having two circles as invariant algebraic limit cycles. The first author is partially supported by a MINECO grant number MTM2014-53703-P, and an AGAUR (Generalitat de Catalunya) grant number 2014SGR 1204. The second author is partially supported by a MINECO grant MTM2013-40998-P, an AGAUR grant 2014SGR 568, and two grants FP7-PEOPLE-2012-IRSES numbers 316338 and 318999. The third author is partially supported by FCT/Portugal through the project UID/MAT/04459/2013.

V članku pokažemo, da je relativna normalizirana velikost algebraičnega celega števila ?$\alpha \ne -1, 0, 1$? glede na številski obseg ?$\mathbb{K}$? večja od 1 pod pogojem, da število realnih vložitev ?$s$? obsega ?$\mathbb{K}$? zadošča pogoju ?$s \ge 0.828n$?, kjer je ?$n = \mathbb{K} : \mathbb{Q}$?. To lahko primerjamo s prejšnjo veliko bolj omejujočo oceno ?$s \ge n - 0.192p \sqrt{n/\log n}$?. To tudi pokaže, da je minimalna vrednost ?$m(\mathbb{K}$? nad relativno normalizirano velikostjo neničelnih algebraičnih celih števil ?$\alpha$? v takšnem obsegu ?$\mathbb{K}$? enaka 1, in ta minimum je dosežen pri ?$\alpha = \pm 1$?. Močnejšo kot prejšnjo, toda verjetno ne optimalno mejo za ?$m(\mathbb{K}$?, dobimo tudi za obsege ?$\mathbb{K}$?, ki zadoščajo pogoju ?$0.639 \le s/n < 0.827469$?.... V dokazu uporabimo spodnjo mejo za Mahlerjevo mero algebraičnega števila z veliko realnimi konjugiranimi števili.

The notions of equivalence and strict equivalence for order one differential equations are introduced. The more explicit notion of strict equivalence is applied to examples and questions concerning autonomous equations and equations having the Painleve property. The order one equation determines an algebraic curve. If this curve has genus zero or one, then it is difficult to verify strict equivalence. However, for higher genus strict equivalence can be tested by an algorithm sketched in the text. For autonomous equations, testing strict equivalence and the existence of algebraic solutions are shown to be algorithmic., 19 pages

The differential equation f ( k ) = f ( j 1 ) … f ( j d ) with d > 1 and each j i k has no entire transcendental solutions. In a sense, in almost all cases transcendental meromorphic solutions can also be excluded, and with substantially fewer possible exceptional cases transcendental solutions are either elliptic or of the form f ( z ) = g ( e c z ) with g rational and c constant.

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)].

Let G be a simple algebraic group defined over an algebraically closed field K and let G = G ( K ) . We consider here the problem when an element g ∈ G can be presented in the form g = e n ( g 1 , g 2 ) : = [ g 1 , [ g 1 , ⋯ [ g 1 ︸ n -times , g 2 ] ⋯ ] ] for some g 1 , g 2 ∈ G . This can always be done if g is a semisimple or a unipotent element. Also, it is known that this holds for G = SL 2 ( K ) . Here we prove that g = e n ( g 1 , g 2 ) for every n if G is a group of type B 2 or G 2 and we prove g = e 2 ( g 1 , g 2 ) = [ g 1 , [ g 1 , g 2 ] ] if G = PGL 3 ( K ) . We also show g = e 2 ( g 1 , g 2 ) = [ g 1 , [ g 1 , g 2 ] ] if g is a regular element of G and G is a group of rank 3. For any simple group G we give a criterion for some “general” regular elements of G to be presented in the form [ g 1 , [ g 1 , g 2 ] ] .

We consider real algebraic numbers alpha of degree deg alpha = n and height H = H(alpha). There are intervals I subset of R of length vertical bar I vertical bar whose interiors contain no real algebraic numbers alpha of any degree with H(alpha) < 1/2 vertical bar I vertical bar(-1). We prove that one can always find a constant c(1) = c(1)(n) such that if Q is a positive integer and Q > c(1)vertical bar I vertical bar(-1), then the interior of I contains at least c(2)(n)Q(n+1) vertical bar I vertical bar real algebraic numbers alpha with deg alpha = n and H(alpha)

Let G be a connected reductive algebraic group defined over an algebraically closed field k. The aim of this paper is to present a method to find triples ( G , M , H ) with the following three properties. Property 1: G is simple and k has characteristic 2. Property 2: H and M are closed reductive subgroups of G such that H M G , and ( G , M ) is a reductive pair. Property 3: H is G-completely reducible, but not M-completely reducible. We exhibit our method by presenting a new example of such a triple in G = E 7 . Then we consider a rationality problem and a problem concerning conjugacy classes as important applications of our construction.

In this paper, we iterate the algebraic etale-Brauer set for any nice variety X over a number field k with π 1 et ( X ‾ ) finite and we show that the iterated set coincides with the original algebraic etale-Brauer set. This provides some evidence towards the conjectures by Colliot-Thelene on the arithmetic of rational points on nice geometrically rationally connected varieties over k and by Skorobogatov on the arithmetic of rational points on K3 surfaces over k; moreover, it gives a partial answer to an “algebraic” analogue of a question by Poonen about iterating the descent set.

Agraïments: The second author is partially supported by FCT/Portugal through UID/MAT/04459/2013. Algebraic limit cycles in quadratic polynomial differential systems started to be studied in 1958, and few years later the following conjecture appeared: Quadratic polynomial differential systems have at most one algebraic limit cycle. We prove that for a quadratic polynomial differential system having two pairs of diametrally opposite equilibrium points at infinity, has at most one algebraic limit cycle. Our result provides a partial positive answer to this conjecture.

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, we study various properties of algebraic extension of *-A operator. Specifically, we show that every algebraic extension of *-A operator has SVEP and is isoloid. And if T is an algebraic extension of *-A operator, then Weyl's theorem holds for f(T), where f is an analytic functions on some neighborhood of σ(T) and not constant on each of the components of its domain.

We use a generalization of a construction by Ziegler to show that for any field F and any countable collection of countable subsets A i ⊆ F , i ∈ I ⊂ Z > 0 there exist infinitely many fields K of arbitrary greater than one transcendence degree over F and of infinite algebraic degree such that each A i is first-order definable over K. We also use the construction to show that many infinitely axiomatizable theories of fields which are not compatible with the theory of algebraically closed fields are finitely hereditarily undecidable.

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.

A method of solving polynomial equations in a ring D[x] is described, where D is an arbitrary order of field ℚ(ω) and ω is an algebraic integer number.

Let k be an arbitrary field, let G be a (smooth) linear algebraic group over k, and let U be a vector group over k on which G acts by automorphisms of algebraic groups. The action of G on U is said to be linear if there is a G-equivariant isomorphism of algebraic groups U ≃ Lie ( U ) . Suppose that G is connected and that the unipotent radical of G is defined over k. If the G-module Lie ( U ) is simple, we show that the action of G on U is linear. If G acts by automorphisms on a connected, split unipotent group U, we deduce that U has a filtration by G-invariant closed subgroups for which the successive factors are vector groups with a linear action of G. When G is connected and the unipotent radical of G is defined and split over k, this verifies an assumption made in earlier work of the author on the existence of Levi factors. On the other hand, for any field k of positive characteristic we show that if the category of representations of G is not semisimple, there is an action of G on a suitable vector group U which is not linear.

In the present paper, we classify varieties of algebraic systems of the type [Formula: see text], for natural numbers [Formula: see text] and [Formula: see text], which are closed under particular derived algebraic systems. If we replace in an algebraic system the [Formula: see text]-ary operation by an [Formula: see text]-ary term operation and the [Formula: see text]-ary relation by the [Formula: see text]-ary relation generated by an [Formula: see text]-ary formula, we obtain a new algebraic system of the same type, which we call derived algebraic system. We shall restrict the replacement to so-called “linear” terms and atomic “linear” formulas, respectively.

For each element a in the Banach algebra A , we define the resolvent space R a and completely characterize it whenever a is algebraic. In particular, we find elements a with R a ≠ { a } ′ . Then we consider the Banach algebra of operators L ( X ) , and show that R A possesses nontrivial invariant subspaces whenever A is an algebraic element of L ( X ) . This assertion becomes stronger than that of the existence of a hyper-invariant subspace for A whenever R A ≠ { A } ′ .

Various aspects of the work of Blok and Rebagliato on the algebraic semantics for deductive systems are studied in the context of logics formalized as π-institutions. Three kinds of semantics are surveyed: institution, matrix (system) and algebraic (system) semantics, corresponding, respectively, to the generalized matrix, matrix and algebraic semantics of the theory of sentential logics. After some connections between matrix and algebraic semantics are revealed, it is shown that every (finitary) N -rule based extension of an N -rule based π-institution possessing an algebraic semantics also possesses an algebraic semantics. This result abstracts one of the main theorems of Blok and Rebagliato. An attempt at a Blok-Rebagliato-style characterization of those π-institutions with a mono-unary category of natural transformations on their sentence functors having an algebraic semantics is also made. Finally, a necessary condition for a π-institution to possess an algebraic semantics is provided. c

Power mapping based S-boxes, especially those with finite field inversion, have received significant attention by cryptographers. S-boxes designed by finite field inversion provide good cryptographic properties and are used in most [email protected]? design such as Advanced Encryption Standard (AES), Camellia, Shark and others. However, such an S-box consists of a simple algebraic expression, thus the S-box design is completed by adding an affine transformation before the input of the S-box, or after the output of the S-box or both in order to make the overall S-box description more complex in a finite field. In the present study, a new method of computation of the algebraic expression (as a polynomial function over GF(2^8)) of power mapping based S-boxes designed by three different probable cases is described in which the place of the affine transformation differs. The proposed method is compared with the Lagrange interpolation formula with respect to the number of polynomial operations needed. The new method (based on the square-and-multiply technique) is found to reduce time and polynomial operation complexity in the computation of the algebraic expression of S-boxes.

Let K be a field of characteristic zero, alpha algebraic of degree n over K. Given a proper parametrization psi of a rational curve C, we present a new algorithm to compute the hypercircle associated to the parametrization psi. As a consequence, we can decide if the curve C is defined over K and, if not, to compute the minimum field of definition of C containing K. The algorithm exploits the conjugate curves of C but avoids computation in the normal closure of K(alpha) over K., 16 pages