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Displaying 2921 – 2940 of 3014

On wild ramification in quaternion extensions

G. Griffith Elder, Jeffrey J. Hooper (2007)

Journal de Théorie des Nombres de Bordeaux

This paper provides a complete catalog of the break numbers that occur in the ramification filtration of fully and thus wildly ramified quaternion extensions of dyadic number fields which contain $\sqrt{- 1}$ (along with some partial results for the more general case). This catalog depends upon the refined ramification filtration, which as defined in [2] is associated with the biquadratic subfield. Moreover we find that quaternion counter-examples to the conclusion of the Hasse-Arf Theorem are extremely rare...

On Witt rings of function fields of real analytic surfaces and curves.

Piotr Jaworski (1997)

Revista Matemática de la Universidad Complutense de Madrid

Let V be a paracompact connected real analytic manifold of dimension 1 or 2, i.e. a smooth curve or surface. We consider it as a subset of some complex analytic manifold VC of the same dimension. Moreover by a prime divisor of V we shall mean the irreducible germ along V of a codimension one subvariety of VC which is an invariant of the complex conjugation. This notion is independent of the choice of the complexification VC. In the one-dimensional case prime divisors are just points, in the two-dimensional...

On Witten multiple zeta-functions associated with semisimple Lie algebras I

Kohji Matsumoto, Hirofumi Tsumura (2006)

Annales de l’institut Fourier

We define Witten multiple zeta-functions associated with semisimple Lie algebras $𝔰𝔩 (n)$ , $(n = 2, 3, ...)$ of several complex variables, and prove the analytic continuation of them. These can be regarded as several variable generalizations of Witten zeta-functions defined by Zagier. In the case $𝔰𝔩 (4)$ , we determine the singularities of this function. Furthermore we prove certain functional relations among this function, the Mordell-Tornheim double zeta-functions and the Riemann zeta-function. Using these relations, we prove...

On $X_{1}^{4} + 4 X_{2}^{4} = X_{3}^{8} + 4 X_{4}^{8}$ and $Y_{1}^{4} = Y_{2}^{4} + Y_{3}^{4} + 4 Y_{4}^{4}$

Susil Kumar Jena (2015)

Communications in Mathematics

The two related Diophantine equations: $X_{1}^{4} + 4 X_{2}^{4} = X_{3}^{8} + 4 X_{4}^{8}$ and $Y_{1}^{4} = Y_{2}^{4} + Y_{3}^{4} + 4 Y_{4}^{4}$ , have infinitely many nontrivial, primitive integral solutions. We give two parametric solutions, one for each of these equations.

On $x^{n} + y^{n} = n! z^{n}$

Susil Kumar Jena (2018)

Communications in Mathematics

In p. 219 of R.K. Guy’s Unsolved Problems in Number Theory, 3rd edn., Springer, New York, 2004, we are asked to prove that the Diophantine equation $x^{n} + y^{n} = n! z^{n}$ has no integer solutions with $n \in ℕ_{+}$ and $n > 2$ . But, contrary to this expectation, we show that for $n = 3$ , this equation has infinitely many primitive integer solutions, i.e. the solutions satisfying the condition $gcd (x, y, z) = 1$ .