197 research outputs found
Partial zeta functions of algebraic varieties over finite fields
By restricting the variables running over various (possibly different)
subfields, we introduce the notion of a partial zeta function. We prove that
the partial zeta function is rational in an interesting case, generalizing
Dwork's well known rationality theorem. In general, the partial zeta function
is probably not rational. But a theorem of Faltings says that the partial zeta
function is always nearly rational
Rationality of partial zeta functions
We prove that the partial zeta function introduced in [9] is a rational
function, generalizing Dwork's rationality theorem.Comment: 8 page
Higher rank case of Dwork's conjecture
This is the final version of ANT-0142 ("An embedding approach to Dwork's
conjecture"). It reduces the higher rank case of the conjecture over a general
base variety to the rank one case over the affine space. The general rank one
case is completed in ANT-0235 "Rank one case of Dwork's conjecture". Both
papers will appear in JAMS
On the subset sum problem over finite fields
The subset sum problem over finite fields is a well-known {\bf NP}-complete
problem. It arises naturally from decoding generalized Reed-Solomon codes. In
this paper, we study the number of solutions of the subset sum problem from a
mathematical point of view. In several interesting cases, we obtain explicit or
asymptotic formulas for the solution number. As a consequence, we obtain some
results on the decoding problem of Reed-Solomon codes.Comment: 16 page
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