Some algebraic and geometric computations in PSPACE
STOC '88 Proceedings of the twentieth annual ACM symposium on Theory of computing
Quantifier elimination in the theory of an algebraically-closed field
STOC '89 Proceedings of the twenty-first annual ACM symposium on Theory of computing
Computational Aspects of Curves of Genus at Least 2
ANTS-II Proceedings of the Second International Symposium on Algorithmic Number Theory
Counting Rational Points on Curves and Abelian Varieties over Finite Fields
ANTS-II Proceedings of the Second International Symposium on Algorithmic Number Theory
The complexity of quantifier elimination in the theory of an algebraically closed field
The complexity of quantifier elimination in the theory of an algebraically closed field
Genus 2 point counting over prime fields
Journal of Symbolic Computation
Counting points on Cab curves using Monsky--Washnitzer cohomology
Finite Fields and Their Applications
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We develop efficient methods for deterministic computations with semi-algebraic sets and apply them to the problem of counting points on curves and Abelian varieties over finite fields. For Abelian varieties of dimension g in projective N space over Fq, we improve Pila's result and show that the problem can be solved in O((q)) time where is polynomial in g as well as in N. For hyper elliptic curves of genus g over Fq we show that the number of rational points on the curve and the number of rational points on its Jacobian can be computed in (q)O(g2g)time.