The algebraic fundamental group of a reductive group scheme over an arbitrary base scheme

We define the algebraic fundamental group π ₁(G) of a reductive group scheme G over an arbitrary non-empty base scheme and show that the resulting functor G? π₁(G) is exact.     

Ideal membership in polynomial rings over the integers

We present a new approach to the ideal membership problem for polynomial rings over the integers: given polynomials , where is an -tuple of indeterminates, are there such that ? We show that the degree of the polynomials can be bounded by where is the maximum total degree and the maximum height of the coefficients of . Some related questions, primarily concerning linear equations in , where is the ring of integers of a number field, are also treated.

     

Approximation of a two-valued function by an algebraic polynomial

We consider the minimax model of a nonlinear structure for approximating a two-valued function by an algebraic polynomial. We establish optimality conditions as a strong generalization of P. L. Chebyshev alternance optimality conditions in approximation of a function by a polynomial.     

Solution of polynomial equations in the field of algebraic numbers

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

SomeL-functions of a polynomial ring over a finite field

An L-function involved in the enumeration of certain families of irreducible polynomials over a finite field is studied. For a number of cases, we confirm the hypothesis on zeros of that function, which is an analog of the hypothesis of Riemann and Weyl postulated for algebraic varieties over finite fields. Translated fromAlgebra i Logika, Vol. 35, No. 4, pp. 476–495, July–August, 1996.     

The fundamental group of an algebraic link

We compute the fundamental group of an algebraic link. To cite this article: O. Neto, P.C. Silva, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

The Geisser-Levine method revisited and algebraic cycles over a finite field

We reformulate part of the arguments of T. Geisser and M. Levine relating motivic cohomology with finite coefficients to truncated étale cohomology with finite coefficients [9,10]. This reformulation amounts to a uniqueness theorem for motivic cohomology, and shows that the Geisser-Levine method can be applied generally to compare motivic cohomology with other types of cohomology theories. We apply this to prove an equivalence between conjectures of Tate and Beilinson on cycles in characteristic p and a vanishing conjecture for continuous étale cohomology. Received: 23 November 2000 / Published online: 5 September 2002     

The Picard group of algebras over an algebraically nonclosed field

Let k be a field that is not algebraically closed, and let A be a k-algebra, whose each maximal ideal has residue field k. We prove that each element of the Picard group of A is of finite order and give an optimal upper bound for its order.     

The trace-form of an algebraic number field

Let F be the rational field or a p-adic field, and let K an algebraic number field over F. If ω₁,…, ω_n is an integral basis for the ring $\text{D}$_L of integers in K, then the quadratic form Q whose matrix is (trace_K∥F(ω_iω_j)) has integral coefficients, and is called an integral trace-form. Q is determined by K up to integral equivalence. The purpose of this paper is to show that the genus of Q determines the ramification of primes in K.     

Harish-Chandra pairs for algebraic affine supergroup schemes over an arbitrary field

Over an arbitrary field of characteristic ≠ 2, we define the notion of Harish-Chandra pairs, and prove that the category of those pairs is anti-equivalent to the category of algebraic affine supergroup schemes. The result is applied to characterize some classes of affine supergroup schemes such as those which are (a) simply connected, (b) unipotent or (c) linearly reductive in positive characteristic.