Norm form equations IV: Rational functions

Parameterized norm form equations with arithmetic progressions

Buchmann and Pethő [5] observed that following algebraic integer 10 + 9α + 8α + 7α + 6α + 5α + 4α, with α = 3 is a unit. Since the coefficients form an arithmetic progressions they have found a solution to the Diophantine equation (1) NK/Q(x0 + αx1 + · · ·+ x6α) = ±1, such that (x0, . . . , x6) ∈ Z is an arithmetic progression. Recently Bérczes and Pethő [3] considered the Diophantine equation ...

Norm Form Equations and Continued Fractions

We consider the Diophantine equation of the form x2−Dy2 = c, where c ∣∣ 2D, gcd(x, y) = 1, and provide criteria for solutions in terms of congruence conditions on the fundamental solution of the Pell Equation x2 − Dy2 = 1. The proofs are elementary, using only basic properties of simple continued fractions. The results generalize various criteria for such solutions, and expose the central norm,...

The Norm Form of a Rational Division Algebra.

Nonexistence of generalized bent functions and the quadratic norm form equations

We obtain the nonexistence of generalized bent functions (GBFs) from (\ZZ/t\ZZ)^n to \ZZ/t\ZZ (called type [n,t]), for a large new class. Specifically, by showing certain quadratic norm form equations have no integral points, we obtain the universal nonexistence of GBFs with type [n, 2p^e] for all sufficiently large p with respect to n and (p-1)/\ord_2(p), and by computational methods with a we...

The best approximation of some rational functions in uniform norm

Here we are concerned with the best approximation by polynomials to rational functions in the uniform norm. We give some new theorems about the best approximation of 1/(1 + x) and 1/(x − a) where a > 1. Finally we extend this problem to that of computing the best approximation of the Chebyshev expansion in uniform norm and give some results and conjectures about this.  2005 IMACS. Published by...