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    Computing the pp-adic Canonical Quadratic Form in Polynomial Time

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    An nn-ary integral quadratic form is a formal expression Q(x1,..,xn)=βˆ‘1≀i,j≀naijxixjQ(x_1,..,x_n)=\sum_{1\leq i,j\leq n}a_{ij}x_ix_j in nn-variables x1,...,xnx_1,...,x_n, where aij=aji∈Za_{ij}=a_{ji} \in \mathbb{Z}. We present a randomized polynomial time algorithm that given a quadratic form Q(x1,...,xn)Q(x_1,...,x_n), a prime pp, and a positive integer kk outputs a U∈GLn(Z/pkZ)\mathtt{U} \in \text{GL}_n(\mathbb{Z}/p^k\mathbb{Z}) such that U\mathtt{U} transforms QQ to its pp-adic canonical form.Comment: arXiv admin note: text overlap with arXiv:1404.028
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