102 research outputs found
On the Number of Places of Convergence for Newton's Method over Number Fields
Let f be a polynomial of degree at least 2 with coefficients in a number
field K, let x_0 be a sufficiently general element of K, and let alpha be a
root of f. We give precise conditions under which Newton iteration, started at
the point x_0, converges v-adically to the root alpha for infinitely many
places v of K. As a corollary we show that if f is irreducible over K of degree
at least 3, then Newton iteration converges v-adically to any given root of f
for infinitely many places v. We also conjecture that the set of places for
which Newton iteration diverges has full density and give some heuristic and
numerical evidence.Comment: 9 pages; minor changes from the previous version; to appear in
Journal de Th\'eorie des Nombres de Bordeau
A note on the arithmetic of differential equations
AbstractIn this note we give a method for computing the differential Galois group of some linear second-order ordinary differential equations using arithmetic information, namely the p-curvatures
On (k,n)-arcs
AbstractWe construct (k,n)-arcs in PG(2,q) with k approximately q2/d and n approximately q/d for each divisor d of q-1
Diophantine approximation and deformation
We associate certain curves over function fields to given algebraic power
series and show that bounds on the rank of Kodaira-Spencer map of this curves
imply bounds on the exponents of the power series, with more generic curves
giving lower exponents. If we transport Vojta's conjecture on height inequality
to finite characteristic by modifying it by adding suitable deformation
theoretic condition, then we see that the numbers giving rise to general curves
approach Roth's bound. We also prove a hierarchy of exponent bounds for
approximation by algebraic quantities of bounded degree
Recovering affine curves over finite fields from -functions
Let be the function field of a curve over a finite field of odd
characteristic. We investigate using -functions of Galois extensions of
to effectively recover . When is the function field of the projective
line with four rational points removed, we show how to use -functions of a
ray class field to effectively recover the removed points up to automorphisms
of the projective line. When is the function field of a plane curve, we
show how to effectively recover the equation of that curve using -functions
of Artin-Schreier extensions of .Comment: 23 pages; comments welcom
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