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Computing Jacobi’s theta in quasi-linear time. (English) Zbl 1430.11167
Summary: Jacobi’s $$\theta$$ function has numerous applications in mathematics and computer science; a naive algorithm allows the computation of $$\theta (z,\tau )$$, for $$z, \tau$$ verifying certain conditions, with precision $$P$$ in $$O(\mathcal {M}(P) \sqrt {P})$$ bit operations, where $$\mathcal {M}(P)$$ denotes the number of operations needed to multiply two complex $$P$$-bit numbers. We generalize an algorithm which computes specific values of the $$\theta$$ function (the theta-constants) in asymptotically faster time; this gives us an algorithm to compute $$\theta (z, \tau )$$ with precision $$P$$ in $$O(\mathcal {M}(P) \log P)$$ bit operations, for any $$\tau \in \mathcal {F}$$ and $$z$$ reduced using the quasi-periodicity of $$\theta$$.

##### MSC:
 11Y16 Number-theoretic algorithms; complexity 14Q20 Effectivity, complexity and computational aspects of algebraic geometry 14Q05 Computational aspects of algebraic curves 14H42 Theta functions and curves; Schottky problem 14K25 Theta functions and abelian varieties 14H81 Relationships between algebraic curves and physics 11-04 Software, source code, etc. for problems pertaining to number theory
MPC
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