Ternary quadratic forms and quaternion algebras
نویسندگان
چکیده
منابع مشابه
Representation by Ternary Quadratic Forms
The problem of determining when an integral quadratic form represents every positive integer has received much attention in recent years, culminating in the 15 and 290 Theorems of Bhargava-Conway-Schneeberger and Bhargava-Hanke. For ternary quadratic forms, there are always local obstructions, but one may ask whether there are ternary quadratic forms which represent every locally represented in...
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We propose an algorithm for finding zero divisors in quaternion algebras over quadratic number fields, or equivalently, solving homogeneous quadratic equations in three variables over Q( √ d) where d is a square-free integer. The algorithm is deterministic and runs in polynomial time if one is allowed to call oracles for factoring integers and polynomials over finite fields.
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We discuss the relationship between quaternion algebras and quadratic forms with a focus on computational aspects. Our basic motivating problem is to determine if a given algebra of rank 4 over a commutative ring R embeds in the 2 × 2-matrix ring M2(R) and, if so, to compute such an embedding. We discuss many variants of this problem, including algorithmic recognition of quaternion algebras amo...
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We show that a positive de nite integral ternary form can be reduced with O(M(s) log s) bit operations, where s is the binary encoding length of the form and M(s) is the bit-complexity of s-bit integer multiplication. This result is achieved in two steps. First we prove that the the classical Gaussian algorithm for ternary form reduction, in the variant of Lagarias, has this worst case running ...
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This paper specifies some conditions as to when an integer m is locally represented by a positive definite diagonal integer-matrix ternary quadratic form Q at a prime p. We use quadratic Gauss sums and a version of Hensel’s Lemma to count how many solutions there are to the equivalence Q(~x) ≡ m (mod p) for any k ≥ 0. Given that m is coprime to the determinant of the Hessian matrix of Q, we can...
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ژورنال
عنوان ژورنال: Journal of Number Theory
سال: 1986
ISSN: 0022-314X
DOI: 10.1016/0022-314x(86)90090-9