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In this paper, we study the polynomial Gauss sums over finite fields, and present an analogue of Davenport–Hasse's theorem for the polynomial Gauss sums, which is a generalization of the previous result obtained by Hayes. [ABSTRACT FROM AUTHOR]

A univariate polynomial f ( x ) is said to be nice if all of its coefficients as well as all of the roots of both f ( x ) and its derivative f ′ ( x ) are integers. The known examples of nice polynomials with distinct roots are limited to quadratic polynomials, cubic polynomials, symmetric quartic polynomials and, up to equivalence, only a finite number of nonsymmetric quartic polynomials and one quintic polynomial. In this paper we find parametrized families of nice nonsymmetric quartic polynomials with distinct roots, as well as infinitely many nice quintic and sextic polynomials with distinct roots. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we determine all sets A of integers such that, for any integral-valued polynomial which has no fixed divisor, for all integers and n, there are infinitely many integers and a choice of such that . The earlier result shows that is such a set. [Copyright &y& Elsevier]

Abstract: Let m be a positive integer and be a polynomial of the form . We call a polynomial a Rabinowitsch polynomial if for and consecutive integers , is either 1 or prime. In this paper, we show that there are exactly 14 Rabinowitsch polynomials . [Copyright &y& Elsevier]

Abstract: In this paper we prove the best possible upper bounds for the number of elements in a set of polynomials with integer coefficients all having the same degree, such that the product of any two of them plus a linear polynomial is a square of a polynomial with integer coefficients. Moreover, we prove that there does not exist a set of more than 12 polynomials with integer coefficients and with the property from above. This significantly improves a recent result of the first two authors with Tichy [A. Dujella, C. Fuchs, R.F. Tichy, Diophantine m-tuples for linear polynomials, Period. Math. Hungar. 45 (2002) 21–33]. [Copyright &y& Elsevier]

Abstract: Let be a -sheaf. Building on previous work of Drinfeld, Anderson, Taguchi, and Wan, Böckle and Pink (A cohomological theory of crystals over function fields, in preparation), develop a cohomology theory for . Böckle (Math. Ann. 323 (2002) 737) uses this theory to establish the analytic continuation of the L-series associated to (which is a characteristic p-valued “Dirichlet series”) and the logarithmic growth of the degrees of its special polynomials. In this paper, we shall show that this logarithmic growth is all that is needed to analytically continue the original L-series as well as all associated partial L-series. Moreover, we show that the degrees of the special polynomials attached to the partial L-series also grow logarithmically. Our tools are Böckle''s original results, non-Archimedean integration, and the very strong estimates of Amice (Bull. Soc. Math. France 92 (1964) 117). Along the way, we define certain natural modules associated with non-Archimedean measures (in the characteristic 0 case as well as in characteristic p). [Copyright &y& Elsevier]

In his 1964 paper, de Bruijn (Math. Comp. 18 (1964) 537) called a pair (a,b) of positive odd integers good, if Z=aS⊖2bS, where S is the set of nonnegative integers whose 4-adic expansion has only 0''s and 1''s, otherwise he called the pair (a,b) bad. Using the 2-adic integers we obtain a characterization of all bad pairs. A positive odd integer u is universally bad if (ua,b) is bad for all pairs of positive odd integers a and b. De Bruijn showed that all positive integers of the form u=2k+1 are universally bad. We apply our characterization of bad pairs to give another proof of this result of de Bruijn, and to show that all integers of the form u=φpk(4) are universally bad, where p is prime and φn(x) is the nth cyclotomic polynomial. We consider a new class of integers we call de Bruijn universally bad integers and obtain a characterization of such positive integers. We apply this characterization to show that the universally bad integers u=φpk(4) are in fact de Bruijn universally bad for all primes p>2. Furthermore, we show that the universally bad integers φ2k(4), and more generally, those of the form 4k+1, are not de Bruijn universally bad. [Copyright &y& Elsevier]