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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.

A new algebraic proof and method for the exact computation of the number of zeros of a real polynomial inside the unit disk is given in this paper.

Let VK be a twisted form of P3 over the function field of an algebraic surface with Pic VK generated by the half of the canonical line bundle. We construct an algebraic fibre space V → X projective flat over a smooth projective surface X with the generic fibre V K → Spac K satisfying some properties.

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.

Suppose that is an algebraic field; if is complex, and if ; . For let , where is the Weil height of the number . Then the inequality holds for multiplicatively independent .

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.

By means of Gröbner basis techniques algorithms for solving various problems concerning subfields K(g):=K(g1, …,gm) of a rational function field K(x):=K(x1, …,xn) are derived: computing canonical generating sets, deciding field membership, computing the degree and separability degree resp. the transcendence degree and a transcendence basis of K(x)/K(g), deciding whetherf∈K(x) is algebraic or transcendental over K(g), computing minimal polynomials, and deciding whether K(g) contains elements of a “particular structure”, e.g. monic univariate polynomials of fixed degree. The essential idea is to reduce these problems to questions concerning an ideal of a polynomial ring; connections between minimal primary decompositions over K(x) of this ideal and intermediate fields of K(g) and K(x) are given. In the last section some practical considerations concerning the use of the algorithms are discussed.

Let K be a quadratic extension of a field k which is either local field or a finite field. Let G be an algebraic group over k. The aim of the present paper is to understand when a representation of G(K) has a G(k) invariant linear form. We are able to accomplish this in the case when G is the group of invertible elements of a division algebra over k of odd index if k is a local field, and for general connected groups over finite fields.

only a finite number of solutions. Unfortunately, the underlying method of Thue–Siegel–Roth is ineffective in the sense that it does not provide upper bounds for y or H0(β) respectively. However, it allows giving an explicit upper bound for the number of x/y ∈ Q satisfying (1.1). A first result was proved by Davenport and Roth ([3], 1955). This bound was improved by Bombieri and van der Poorten ([1], 1987) and independently by Luckhardt ([10], 1989) using the modified proof

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.

Algebraic independence of the numbers where {Rn}n≥0 is a sequence of integers satisfying a binary linear recurrence relation and {bh}h≥0 is a periodic sequence of algebraic numbers not identically zero, are studied.

Let K be a commutative field of characteristic 0, and a an algebraic series of one variable with coefficients in K, which is invertible for the shuffle product, i.e. it exists an algebraic series b over K such that a@?b==1. Then a is a rational series. The same result is true when a and b are an algebraic series of many non-commutative variables. One application of this result is: If M is a linear system which is invertible, and with an output function ) and and N is a linear system with an output function ) suc such that N and the gen and the generating series H for M and G for N are both algebraic, then the systems M and N are bilinear.

We compute the two-completed algebraic K-groups K ∗ ( Z 2 ) 2 ∧ of the two-adic integers, and determine the homotopy type of the two-completed algebraic K-theory spectrum K( Z 2 ) 2 ∧ . The natural map K(Z) 2 ∧ → K( Z 2 ) 2 ∧ is shown to induce an isomorphism modulo torsion in degrees 4k + 1 with k ≥ 1.

We prove that there exists a polynomial of small height with a root close to 1. This implies that there are algebraic numbers close to 1 with relatively small Mahler measure. We also give an explicit construction of such numbers with small Weil height.

Every nonsingular projective real algebraic curve C has a unique, up to isomorphism over ℝ, nonsingular projective complexification V . If V \ C is disconnected, then C is said to be dividing (we identify C with the set of real points of V ). Classical results supply numerous examples of dividing real curves. This class of curves was first studied by Felix Klein [ 11 ]. A characterization of dividing real curves is given in [ 6 ]. In higher dimensions the situation is more complicated. First of all, every nonsingular projective real algebraic variety X of dimension d always has several nonisomorphic nonsingular projective complexifications, provided that d [ges ]2. Furthermore, if d [ges ]2 and W is a nonsingular projective complexification of X , then W \ X is connected (we identify X with the set of real points of W ). What does it then mean for X to be dividing? An answer can be given in terms of homology theory. Let K be a principal ideal domain. Assume that X , regarded as a topological manifold, is orientable over K . We say that X is dividing over K if there exists a fundamental homology class of X over K whose image in H d ( W , K ) under the homomorphism induced by the inclusion map X [rarrhk ] W is zero. In the present paper we prove that this definition does not depend on the choice of W , and give a characterization of real varieties dividing over ℚ. For more information on real varieties dividing over ℤ/2 the reader may consult [ 18 ] (please note that terminology in [ 18 ] is different than here). We also discuss the relationship between real varieties dividing over ℚ (or ℤ) and dividing over ℤ/2. It follows from well-known facts that a real curve is dividing over K if and only if it is dividing.

For real planar polynomial differential systems there appeared a simple version of the $16$th Hilbert problem on algebraic limit cycles: {\it Is there an upper bound on the number of algebraic limit cycles of all polynomial vector fields of degree $m$?} In [J. Differential Equations, 248(2010), 1401--1409] Llibre, Ram\'irez and Sadovskia solved the problem, providing an exact upper bound, in the case of invariant algebraic curves generic for the vector fields, and they posed the following conjecture: {\it Is $1+(m-1)(m-2)/2$ the maximal number of algebraic limit cycles that a polynomial vector field of degree $m$ can have?} In this paper we will prove this conjecture for planar polynomial vector fields having only nodal invariant algebraic curves. This result includes the Llibre {\it et al}\,'s as a special one. For the polynomial vector fields having only non--dicritical invariant algebraic curves we answer the simple version of the 16th Hilbert problem., Comment: 16. Journal Differential Equations, 2011

We characterize the algebraic invariant curves for the Riccati polynomial differential systems of the form x' = 1, y' = a(x)y2 +b(x)y +c(x), where a(x), b(x) and c(x) are arbitrary polynomials. We also characterize their algebraic first integrals.

Let $k$ be a field. We characterize the group schemes $G$ over $k$, not necessarily affine, such that $\mathsf{D}_{\mathrm{qc}}(B_kG)$ is compactly generated. We also describe the algebraic stacks that have finite cohomological dimension in terms of their stabilizer groups., Comment: 17 pages; generalized Theorems A and C; final version; some minor corrections and updated citations

Let L be a recursive algebraic extension of ℚ. Assume that, given α ∈ L, we can compute the roots in L of its minimal polynomial over ℚ and we can determine which roots are Aut(L)-conjugate to α. We prove that there exists a pair of polynomials that characterizes the Aut(L)-conjugates of α, and that these polynomials can be effectively computed. Assume furthermore that L can be embedded in ℝ, or in a finite extension of ℚp (with p an odd prime). Then we show that subsets of L[X]k that are recursively enumerable for every recursive presentation of L[X], are diophantine over L[X].

An estimate of the -norm, , of an arbitrary algebraic polynomial of degree in terms of its values at nodes of a uniform grid is obtained. This estimate shows, in particular, that for with the -norm of a polynomial grows as not faster than the -means, , of this polynomial over the nodes of the grid times some power of .

For an algebraic group R acting morphically on an algebraic variety X the modality of the action, mod(R : X), is the maximal number of parameters on which a family of R-orbits on X depends upon. Let G be a simple algebraic group defined over an algebraically closed field K of characteristic 0. Let P be a parabolic subgroup of G. Then P acts on its unipotent radical Pu via conjugation. The modality of P is defined as mod P ≔ mod(P : Pu). Let r and s be the semisimple rank of G and P respectively. We show that there is a quadratic polynomial ƒ with rational coefficients such that the modality of P is at least ƒ(r − s). In particular, the modality of a Borel subgroup B of G grows at least quadratically with r. As a consequence, we obtain a finiteness result for algebraic groups from [8]: there is only a finite number of simple algebraic groups admitting parabolic subgroups of prescribed semisimple rank and prescribed modality. Combining our lower bounds with upper bounds from [6], we can compute the modality of Borel subgroups in some small rank cases.

Let F(X, Y) be an absolutely irreducible polynomial with coefficients in an algebraic number field K. Denote by C the algebraic curve defined by the equation F(X, Y) = 0 and by K[C] the ring of regular functions on Cover K. Assume that there is a unit ϕ in K[C] − K such that 1 − ϕ is also a unit. Then we establish an explicit upper bound for the size of integral solutions of the equation F(X, Y) = 0, defined over K. Using this result we establish improved explicit upper bounds on the size of integral solutions to the equations defining non-singular affine curves of genus zero, with at least three points at ‘infinity’, the elliptic equations and a class of equations containing the Thue curves.

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).

Let X be a complete normal algebraic variety defined over a field k algebraically closed and let D be a Cartier divisor on X. Under the assumption that D is nef and good, we describe boundedness properties of base locus of the systems |nD|, algebraic convexity properties of X \ supp(D), and a criterium for semiampleness.

In this paper we prove that every real G vector bundles over G circles or on effective G surfaces can be realized by strongly algebraic G vector bundles for finite Abelian groups G . Using this result we prove that every closed orientable smooth three dimensional G manifold is G diffeomorphic to a nonsingular real algebraic G variety for any finite Abelian group G . We also prove that for any finite group G the algebraic realization of smooth G vector bundles over effective G surfaces can be reduced to the algebraic realization of smooth G vector bundles over G circles.

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.

In this paper an algorithm is described for the computation of the dimensions of all irreducible components of an algebraic variety over a zero characteristic ground field. The variety is given as a set of zeros of a family of polynomials of degree less than d in n variables. The working time of the algorithm is polynomial in the size of the input and d n .

It is well known that we can efficiently test whether a polynomial is identically zero or not by examining the values of the polynomial at well-chosen points. Both deterministic and efficient probabilistic algorithms have been devised for this purpose. It is not so well recognized that algebraic functions can be similarly tested for zeroness. The need for zero testing black boxes representing algebraic functions has recently arisen in the area of self-testing/self-correcting programs. Given a black box B α that represents an algebraic function α and a few additional parameters about α, we show how to test if α is equal to the zero function.

We investigate under which conditions an algebraic group G defined over a locally compact field k admitr a subgroup Γ⊂ G(k) which is dense in the Zariski topology, but discerte in the topology induced by the locally compact topology on k. For non—solvable groups we provide a complete answer.

If T is an algebraic torus defined over a discretely valued field K with perfect residue field k, we relate the K-cohomology of T to the k-cohomology of certain objects associated to T. When k has cohomological dimension, Version submitted for publication

The algebraic unknotting number u_a(K) of a knot K was introduced by Hitoshi Murakami. It equals the minimal number of crossing changes needed to turn K into an Alexander polynomial one knot. In a previous paper the authors used the Blanchfield form of a knot K to define an invariant n(K) and proved that n(K) is a lower bound on u_a(K). They also showed that n(K) subsumes all previous classical lower bounds on the (algebraic) unknotting number. In this paper we prove that n(K)=u_a(K)., 32 pages, 18 figures

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.

Let v 0 be a valuation of a field K 0 with value group G 0 . Let K be a function field of a conic over K 0 , and let v be an extension of v 0 to K with value group G such that G/G 0 is not a torsion group. Suppose that either (K 0 , v 0 ) is henselian or v 0 is of rank 1, the algebraic closure of K 0 in K is a purely inseparable extension of K 0 , and G 0 is a cofinal subset of G. In this paper, it is proved that there exists an explicitly constructible element t in K, with v(t) non-torsion modulo G 0 such that the valuation of K 0 (t), obtained by restricting v, has a unique extension to K. This generalizes the result proved by Khanduja in the particular case, when K is a simple transcendental extension of K 0 (compare [4]). The above result is an analogue of a result of Polzin proved for residually transcendental extensions [8].

We show that Diophantine problem (otherwise known as Hilbert's Tenth Problem) is undecidable over the fields of algebraic functions over the finite fields of constants of characteristic greater than two. This is the first example of Diophantine undecidability over any algebraic field. We also show that the Diophantine class of a holomorphy ring of an above mentioned algebraic function field does not change if the set of primes at which the functions of the ring are allowed to have poles is changed by adding or removing of finitely many primes.

The author extend some results obtained by K Gyδry and I. Gaal for number fields and algebraic function fields of characteristic 0, to the fields of algebraic functions of positive characteristic. Though characteristic 0 results in their original form are not true for positive characteristic, one can still effectively classify polynomials with a given discriminant over the fields of algebraic functions of positive characteristic.

We prove that a formal power series in 1/x, whose coefficients are in a field extension of Q and are algebraically independent over Q, is differentially transcendental (i.e. not differentially algebraic) over this field extension. This is stated without proof in [2]. This result provides a source of functions analytic at ∞ that are not differentially algebraic over R. Such functions are of particular interest, because their germs belong to Hardy fields, but not to the class E of [1]-the intersection of all maximal Hardy fields.

This paper is devoted to studying nilpotent algebraic groups (by means of the Putcha-Renner theory of algebraic monoids) and related solvable algebraic monoids defined over an algebraically closed field. The main results are as follow:. 1). If G is a connected algebraic group of dimension ¦ 2, then G is nilpotent iff all the connected closed subgroups of codimension 1 are so. 2). Let G be a connected solvable algebraic group with dim G > 2. If dim G u = 2 and C(G) c ≅G a , then G is nonnilpotent and G u ≅ (G a )2. 3). Let M be a three-dimensional regular nonnilpotent solvable algebraic monoid of rank 1 and M ¦ G(M) = G. If G u ≇ (G a ) 2, then M = G∪E(M) and E(M) \ {1} is an (irreducible closed two-dimensional) subsemigroup, which is either a left zero semigroup or a right zero semigroup.

In 2013, Tang, Carlet, and Tang [IEEE TIT 59(1): 653–664, 2013] presented two classes of Boolean functions. The functions in the first class are unbalanced and the functions in the second one are balanced. Both of those two classes of functions have high nonlinearity, high algebraic degree, optimal algebraic immunity, and high fast algebraic immunity. However, they are not 1-resilient which represents a drawback for their use as filter functions in stream ciphers. In this paper, we first propose a large family of 1-resilient Boolean functions having high lower bound on nonlinearity, optimal algebraic immunity, and optimal algebraic degree, that is, meeting the Siegenthaler bound. Most notably, we can mathematically prove that every function in $n$ variables belonging to this family has fast algebraic immunity no less than $n-6$ , which is the first time that an infinite family of 1-resilient functions with provably high fast algebraic immunity has been invented. Furthermore, we exhibit a subclass of the family which has higher lower bound on nonlinearity than all the known 1-resilient functions with (potentially) optimal algebraic immunity and potentially high fast algebraic immunity.

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.