49,897 research outputs found
On a conjecture on exponential Diophantine equations
We study the solutions of a Diophantine equation of the form ,
where , and . The main
result is that if there exists a solution with odd then
this is the only solution in integers greater than 1, with the possible
exception of finitely many values . We also prove the uniqueness of such
a solution if any of , , is a prime power. In a different vein, we
obtain various inequalities that must be satisfied by the components of a
putative second solution
The Lefschetz-Hopf theorem and axioms for the Lefschetz number
The reduced Lefschetz number, that is, the Lefschetz number minus 1, is
proved to be the unique integer-valued function L on selfmaps of compact
polyhedra which is constant on homotopy classes such that (1) L(fg) = L(gf),
for f:X -->Y and g:Y -->X; (2) if (f_1, f_2, f_3) is a map of a cofiber
sequence into itself, then L(f_2) = L(f_1) + L(f_3); (3) L(f) = - (degree(p_1 f
e_1) + ... + degree(p_k f e_k)), where f is a map of a wedge of k circles, e_r
is the inclusion of a circle into the rth summand and p_r is the projection
onto the rth summand. If f:X -->X is a selfmap of a polyhedron and I(f) is the
fixed point index of f on all of X, then we show that I minus 1 satisfies the
above axioms. This gives a new proof of the Normalization Theorem: If f:X -->X
is a selfmap of a polyhedron, then I(f) equals the Lefschetz number of f. This
result is equivalent to the Lefschetz-Hopf Theorem: If f: X -->X is a selfmap
of a finite simplicial complex with a finite number of fixed points, each lying
in a maximal simplex, then the Lefschetz number of f is the sum of the indices
of all the fixed points of f.Comment: 9 page
On Termination of Integer Linear Loops
A fundamental problem in program verification concerns the termination of
simple linear loops of the form x := u ; while Bx >= b do {x := Ax + a} where x
is a vector of variables, u, a, and c are integer vectors, and A and B are
integer matrices. Assuming the matrix A is diagonalisable, we give a decision
procedure for the problem of whether, for all initial integer vectors u, such a
loop terminates. The correctness of our algorithm relies on sophisticated tools
from algebraic and analytic number theory, Diophantine geometry, and real
algebraic geometry. To the best of our knowledge, this is the first substantial
advance on a 10-year-old open problem of Tiwari (2004) and Braverman (2006).Comment: Accepted to SODA1
Mod pq Galois representations and Serre's conjecture
Motives and automorphic forms of arithmetic type give rise to Galois
representations that occur in {\it compatible families}. These compatible
families are of p-adic representations with p varying. By reducing such a
family mod p one obtains compatible families of mod p representations. While
the representations that occur in such a p-adic or mod p family are strongly
correlated, in a sense each member of the family reveals a new face of the
motive. In recent celebrated work of Wiles playing off a pair of Galois
representations in different characteristics has been crucial.
In this paper we investigate when a pair of mod p and mod q representations
of the absolute Galois group of a number field K simultaneously arises from an
{\it automorphic motive}: we do this in the 1-dimensional (Section 2) and
2-dimensional (Section 3: this time assuming ) cases. In Section
3 we formulate a mod pq version of Serre's conjecture refining in part a
question of Barry Mazur and Ken Ribet.Comment: This is an older preprint that was made available elsewhere on Sep.
19, 200
Sheaves on Toric Varieties for Physics
In this paper we give an inherently toric description of a special class of
sheaves (known as equivariant sheaves) over toric varieties, due in part to A.
A. Klyachko. We apply this technology to heterotic compactifications, in
particular to the (0,2) models of Distler, Kachru, and also discuss how
knowledge of equivariant sheaves can be used to reconstruct information about
an entire moduli space of sheaves. Many results relevant to heterotic
compactifications previously known only to mathematicians are collected here --
for example, results concerning whether the restriction of a stable sheaf to a
Calabi-Yau hypersurface remains stable are stated. We also describe
substructure in the Kahler cone, in which moduli spaces of sheaves are
independent of Kahler class only within any one subcone. We study F theory
compactifications in light of this fact, and also discuss how it can be seen in
the context of equivariant sheaves on toric varieties. Finally we briefly
speculate on the application of these results to (0,2) mirror symmetry.Comment: 83 pages, LaTeX, 4 figures, must run LaTeX 3 times, numerous minor
cosmetic upgrade
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