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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 $M$.

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

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.

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)

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.

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 } ′ .

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

Inspired by the previous work of Tu and Deng, we propose two infinite classes of Boolean functions of 2k variables where k ≥ 2. The first class contains unbalanced functions having high algebraic degree and nonlinearity. The functions in the second one are balanced and have maximal algebraic degree and high nonlinearity (as shown by a lower bound that we prove; as a byproduct we also prove a better lower bound on the nonlinearity of the Carlet-Feng function). Thanks to a combinatorial fact, first conjectured by the authors and later proved by Cohen and Flori, we are able to show that they both possess optimal algebraic immunity. It is also checked that, at least for numbers of variables n ≤ 16, functions in both classes have a good behavior against fast algebraic attacks. Compared with the known Boolean functions resisting algebraic attacks and fast algebraic attacks, both of them possess the highest lower bounds on nonlinearity. These bounds are however not enough for ensuring a sufficient nonlinearity for allowing resistance to fast correlation attack. Nevertheless, as for previously found functions with the same features, there is a gap between the bound that we can prove and the actual values computed for bounded numbers of variables (n ≤ 38). Moreover, these values are very good. The infinite class of functions we propose in Construction 2 presents, among all currently known constructions, the best provable tradeoff between all the important cryptographic criteria.

The template-based method is one of the most successful approaches to algebraic invariant synthesis. In this method, an algorithm designates a template polynomial \(p\) over program variables, generates constraints for \(p=0\) to be an invariant, and solves the generated constraints. However, this approach often suffers from an increasing template size if the degree of a template polynomial is too high.

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.

The purpose of this paper is to define derived schemes over F 1 , the “field with one element”. More generally, we define our derived algebraic geometry over a symmetric monoidal category ( C , ⊗ , 1 ) by adapting the constructions of Lurie. We also suggest that the construction of derived schemes over F 1 has an analogue in the “homotopy theory of algebraic varieties” over an algebraically closed field k , which we refer to as “derived schemes over T 1 ( k ) ”. This is motivated by the fact that the S -modules in the category of schemes over k are analogous to Z -modules.

Let ξ1, ξ2, · · · , ξn be a sequence of mutually independent, identically distributed random variables following semi-stable distribution with characteristic function exp(−(C + cos log|t|)|t|), C > 1. In this work we obtain the lower bound of the number of real zeros of the random algebraic equations n ∑ r=0 arξr(ω)x. This lower bound is Nn > μ( n log n) for all n > n0 outside a set of measure less than μ k0 + 21 2α+1(C+1) α+1 De A ( kn tn )α 1 k0 where n0 ∈ N, ( n) is a sequence such that as n→∞, n log n→∞, but n log n→∞, and 0 < α ≤ 2. −−−−−−−−−−−−−−−−−−−−−−−−−−−−

Based on Tu-Deng's conjecture and the Tu-Deng function, in 2010, X. Tang et al. proposed a class of Boolean functions in even variables with optimal algebraic degree, very high nonlinearity and optimal algebraic immunity. In this corresponding, we consider the concatenation of Tang's function and another Boolean function, and study its cryptographic properties. With this idea, we propose a class of 1-resilient Boolean functions in odd variables with optimal algebraic degree, good nonlinearity and suboptimal algebraic immunity based on Tu-Deng's conjecture.

In this paper, three constructions of balanced Boolean functions with optimal algebraic immunity are proposed. It is checked that, at least for small numbers of input variables, these functions have good behavior against fast algebraic attacks as well. Other cryptographic properties such as algebraic degree and nonlinearity of the constructed functions are also analyzed. Lower bounds on the nonlinearity are proved, which are similar to the best bounds obtained for known Boolean functions resisting algebraic attacks and fast algebraic attacks. Moreover, it is checked that for the number n of variables with 5 ≤ n ≤ 19, the proposed n-variable Boolean functions have in fact very good nonlinearity.

We prove a Darboux–Jouanolou type theorem on the algebraic integrability of polynomial differential r -forms.

In this paper, enlightened by the idea of the weight of a polynomial introduced by Swinnerton-Dyer [2002], we find all the invariant algebraic surfaces of the Chen system x′ = a(y - x), y′ = (c - a)x + cy - xz, z′ = xy - bz.

An (affine) algebraic monoid is an affine variety over an algebraically closed field K endowed with a monoid structure such that the product map is an algebraic variety morphism. Let M be an irreducible algebraic monoid, G its unit group, P a parabolic subgroup of G, and e ∈ M a minimal idempotent. We show that P = C P ( e ) R u ( G ) and that the assignment P ↦ C P ( e ) defines a one-to-one correspondence between parabolic subgroups of G and of C G ( e ) .

Recall that an algebraic module is a K G -module that satisfies a polynomial with integer coefficients, with addition and multiplication given by the direct sum and tensor product. In this article we prove that non-periodic algebraic modules are very rare, and that if the complexity of an algebraic module is at least 3, then it is the only algebraic module on its component of the (stable) Auslander–Reiten quiver. For dihedral 2-groups, we also show that there is at most one algebraic module on each component of the (stable) Auslander–Reiten quiver. We include a strong conjecture on the relationship between periodicity and algebraicity.

In this paper, we explicitly construct a large class of symmetric Boolean functions on 2k variables with algebraic immunity not less than d, where integer k is given arbitrarily and d is a given suffix of k in binary representation. If let d=k, our constructed functions achieve the maximum algebraic immunity. Remarkably, 2⌊log 2k⌋+2 symmetric Boolean functions on 2k variables with maximum algebraic immunity are constructed, which are much more than the previous constructions. Based on our construction, a lower bound of symmetric Boolean functions with algebraic immunity not less than d is derived, which is 2⌊ log 2d⌋+2(k-d+1). As far as we know, this is the first lower bound of this kind.

In this paper, we characterize the set of all ternary algebraic (or polynomial) operations of idempotent algebras that have at least one binary and one ternary algebraic operation depending on every variable, and there exists an integer r > 2 such that there is not any r-ary algebraic operation depending on every variable. We prove that this set forms a finite DeMorgan algebra with a fixed element and then we characterize this DeMorgan algebra.

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

In the context of 2-summable Fredholm modules, we prove that the Connes-Karoubi multiplicative character coincides with Brown's determinant invariant on algebraic K-theory.

A theorem of N. Katz \cite{Ka} p.45, states that an irreducible differential operator $L$ over a suitable differential field $k$, which has an isotypical decomposition over the algebraic closure of $k$, is a tensor product $L=M\otimes_k N$ of an absolutely irreducible operator $M$ over $k$ and an irreducible operator $N$ over $k$ having a finite differential Galois group. Using the existence of the tensor decomposition $L=M\otimes N$, an algorithm is given in \cite{C-W}, which computes an absolutely irreducible factor $F$ of $L$ over a finite extension of $k$. Here, an algorithmic approach to finding $M$ and $N$ is given, based on the knowledge of $F$. This involves a subtle descent problem for differential operators which can be solved for explicit differential fields $k$ which are $C_1$-fields., 21 pages

This dissertation is concerned with the study of irreducible embeddings of simple algebraic groups of exceptional type. It is motivated by the role of such embeddings in the study of positive dimensional closed subgroups of classical algebraic groups. The classification of the maximal closed connected subgroups of simple algebraic groups was carried out by E. B. Dynkin, G. M. Seitz and D. M. Testerman. Their analysis for the classical groups was based primarily on a striking result: if G is a simple algebraic group and o : G → SL(V ) is a tensor indecomposable irreducible rational representation then, with specified exceptions, the image of G is maximal among closed connected subgroups of one of the classical groups SL(V), Sp(V ) or SO(V ). In the case of closed, not necessarily connected, subgroups of the classical groups, one is interested in considering irreducible embeddings of simple algebraic groups and their automorphism groups: given a simple algebraic group Y defined over an algebraically closed field K, one is led to study the embeddings G 0, G is a closed non-connected positive dimensional subgroup of Y and V is a nontrivial irreducible rational KY -module such that V|G is irreducible. We obtain a precise description of such triples (G, Y, V ).

We generalize results of P. Schneider and U. Stuhler for G L l + 1 to a reductive algebraic group G defined and split over a non-Archimedean local field K. Following their lines, we prove that the generalized Steinberg representations of G with coefficients in an arbitrary ring are cyclic. When G is semi-simple of adjoint type, we give an expression of these representations, whenever it is possible and in particular for those that are of maximal degree, in terms of the parahoric subgroups of G.

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.

The main result of this paper is a generalization of the Mezei-Wright theorem, a result on solutions of a system of fixed point equations. In the typical setting, one solves a system of fixed point equations in an algebra equipped with a suitable partial order; there is a least element, suprema of @w-chains exist, the operations preserve the ordering and least upper bounds of @w-chains. In this setting, one solution of this kind of system is provided by least fixed points. The Mezei-Wright theorem asserts that such a solution is preserved by a continuous, order preserving algebra homomorphism. In several settings such as (countable) words or synchronization trees there is no well-defined partial order but one can naturally introduce a category by considering morphisms between the elements. The generalization of this paper consists in replacing ordered algebras by ''categorical algebras''; the least element is replaced by an initial element, and suprema of @w-chains are replaced by colimits of @w-diagrams. Then the Mezei-Wright theorem for categorical algebras is that initial solutions are preserved by continuous morphisms. We establish this result for initial solutions of parametric fixed point equations. One use of the theorem is to characterize an ''algebraic'' element as one that can arise as a solution of some system of fixed point equations. In familiar examples, an algebraic element is one that is context-free, regular or rational. Then, if h:A->B is a continuous morphism of categorical algebras, the algebraic objects in B are those isomorphic to h-images of algebraic objects in A.

We give a uniform method for the two problems of counting the connected and irreducible components of complex algebraic varieties. Our algorithms are purely algebraic, i.e., they use only the field structure of C. They work in parallel polynomial time, i.e., they can be implemented by algebraic circuits of polynomial depth. The design of our algorithms relies on the concept of algebraic differential forms. A further important building block is an algorithm of Szántó computing a variant of characteristic sets. Furthermore, we use these methods to obtain a parallel polynomial time algorithm for computing the Hilbert polynomial of a projective variety which is arithmetically Cohen–Macaulay.

We investigate how well complex algebraic numbers can be approximated by algebraic numbers of degree at most n. We also investigate how well complex algebraic numbers can be approximated by algebraic integers of degree at most n+1. It follows from our investigations that for every positive integer n there are complex algebraic numbers of degree larger than n that are better approximable by algebraic numbers of degree at most n than almost all complex numbers. As it turns out, these numbers are more badly approximable by algebraic integers of degree at most n+1 than almost all complex numbers., Comment: 34 pages

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].