On uniform distribution of sequences inGF {q,x } and [GFq,x]

Summary Analogs are proved for sequences in Φ=GF[q, x] and Φ′=GF {q, x } of results proved in 1962 by C.L. Vanden Eynden concerning uuiform distribution of sequence of integers related to sequences of real numbers. The concept of uniform distribution (mod m), m an integer, in Vanden Eynden's work is sometimes replaced here by modified forms of uniform distribution (mod M) M ∈ Φ. Supported by NSF Research Grant GP 6515. Entrata in Redazione il 13 giugno 1969.     

Uniformly distributed functions inGF[q,x] andGF{q,x}

Summary Let A, B, C, ... denote polynomials over the finite field GF(q). It is shown that the sequence {B_i} is uniformly distributed modulo M if the sequence {B_i+k - B_i} is uniformly distributed modulo M for all integers k>0. A similar result holds for sequences defined by functional values. Also, a result of Weyl concerning uniform distribution modulo 1 is extended to polynomials over finite fields. Entrata in Redazione il 25 febbraio 1972.     

Arithmetic functions associated with unitary divisors inGF[q,x] I

Summary The unitary divisor concept is extended to GF[q, x] and the properties of some elementary functions are obtained. Entrata in Redazione il 9 ottobre 1969.     

[2] x [3] x N is not a circle order

The result stated in the title is proved in this note. Actually we show that S x N is not a circle order, where S={(1, 1), (1, 2), (1, 3), (2, 1), (2, 3)}. Furthermore this non-circle order is critical in the sense that (S-{x}) x N is a circle order for any x in S.     

Homogeneous ideals inK[x,y, z]

Acta Mathematica Hungarica -     

TheT-ideal generated by the standard identitys 3[x 1,x 2,x 3]

LetK = T_o(s₃), {c_n} its codimensions, {l_n} its colengths and {Χ_n} its sequence of co-characters. For 9≦n, c_n =2n - 1 or c_n =n(n + l)/2- 1, 3≦l_n ≦4 and χ_n =[n] + 2[n-1,1] + α[n-2,2] + β[2²,1^n?4] where α + β≦l.     

A note on theT-ideal generated bys 3[x 1,x 2,x 3]

The co-characters of theT-ideal generated by the standard identitys ₃[x ₁,x ₂,x ₃] are determined.     

Some Simple Derivations of k[x,y]

It is shown that the derivation y ^r ? _x  + (xy ^s  + g)? _y , where 0 ≤ r < s are integers, is a simple derivation of k[x, y], the polynomial ring in two variables over a field k of characteristic zero.     

Dedekind subrings ofk[x 1,…,x n ] are rings of polynomials

The purpose of this note is to prove that a Dedekind domain R which contains a field k, and which is a subring ofk[x ₁,…,x _n ] is a ring of polynomials. This generalizes similar results of A. Evyatar and A. Zaks on principal ideal domains, and of P. M. Cohn for the casen=1. Our methods and proofs differ from those introduced previously. This research was partially supported by the National Science Foundation, Grant GP-23861.     

InfiniteB 2[g] sequences

We exhibit, for any integerg≥2, an infinite sequenceA ∈B ₂[g] such that . Furthermore, we obtain better estimates for small values ofg. For instance, we exhibit an infinite sequenceA ∈B ₂[2] such that Partially supported by Colciencias, Colombia and Universidad del Cauca.     

Simplicity of Some Derivations of k[x,y]

If k is a field of characteristic zero, c ∈ k?0, and 1 ≤ s ≤ r are integers such that either r ? s + 1 divides s or s divides r ? s + 1, then it is shown that the derivation y ^r ? _x  + (xy ^s  + c)? _y of the polynomial ring k[x, y] is simple.     

Gauss sums forF q [T]

The purpose of this paper is to define Gauss sums taking values in function fields of one variable over a finite field and to prove analogues of various classical and recent results. These results include Stickelberger's theorem, the Hasse-Davenport theorem, Weil's theorem on Jacobi sums as Hecke characters and the Gross-Koblitz theorem. For comparison, the reader may consult [G-K] and references given there.In this paper we deal only with the simplest case, where the base ringA is the polynomial ringF _q [T] and where we use the Carlitz module; i.e., the simplest rank one Drinfeld module. (See section I). The general case, which has a quite different flavour, will be presented elsewhere. These results formed a part of the author's thesis, Gamma functions and Gauss sums for function fields and periods of Drinfeld modules (Harvard 1987). But the new presentation here is due to a suggestion by Professor Tate. It is my pleasure to thank him.Supported in part by NSF grant DMS 8610730C2     

Einige Anwendungen der Function [x]

Ohne Zusammenfassung