Computing the $p$-adic Canonical Quadratic Form in Polynomial Time
نویسندگان
چکیده
An n-ary integral quadratic form is a formal expression Q(x1, · · · , xn) = ∑ 1≤i,j≤n aijxixj in nvariables x1, · · · , xn, where aij = aji ∈ Z. We present a randomized polynomial time algorithm that given a quadratic form Q(x1, · · · , xn), a prime p, and a positive integer k outputs a U ∈ GLn(Z/p Z) such that U transforms Q to its p-adic canonical form.
منابع مشابه
Determination of a Matrix Function in the Form of f(A)=g(q(A)) Where g(x) Is a Transcendental Function and q(x) Is a Polynomial Function of Large Degree Using the Minimal Polynomial
Matrix functions are used in many areas of linear algebra and arise in numerical applications in science and engineering. In this paper, we introduce an effective approach for determining matrix function f(A)=g(q(A)) of a square matrix A, where q is a polynomial function from a degree of m and also function g can be a transcendental function. Computing a matrix function f(A) will be time- consu...
متن کاملA p-adic algorithm to compute the Hilbert class polynomial
Classicaly, the Hilbert class polynomial P∆ ∈ Z[X] of an imaginary quadratic discriminant ∆ is computed using complex analytic techniques. In 2002, Couveignes and Henocq [5] suggested a p-adic algorithm to compute P∆. Unlike the complex analytic method, it does not suffer from problems caused by rounding errors. In this paper we complete the outline given in [5] and we prove that, if the Genera...
متن کاملEfficient Calculation of Stark-heegner Points via Overconvergent Modular Symbols
This note presents a qualitative improvement to the algorithm presented in [DG] for computing Stark-Heegner points attached to an elliptic curve and a real quadratic field. This algorithm computes the Stark-Heegner point with a p-adic accuracy of M significant digits in t ime which is polynomial in M, the prime p being treated as a constant, rather than the O(p M) operations required for the mo...
متن کاملFactorization of Quadratic Polynomials in the Ring of Formal Power Series over Z
We establish necessary and sufficient conditions for a quadratic polynomial to be irre-ducible in the ring Z[[x]] of formal power series over the integers. In particular, for polynomials of the form p n + p m βx + αx 2 with n, m ≥ 1 and p prime, we show that reducibility in Z[[x]] is equivalent to reducibility in Zp[x], the ring of polynomials over the p-adic integers.
متن کاملOF THE p - ADIC QUADRATIC RESIDUE CODES Sung
Using the Newton’s identities, we give the inductive formula for the generator polynomials of the p-adic quadratic residue codes.
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید
ثبت ناماگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید
ورودعنوان ژورنال:
- CoRR
دوره abs/1409.6199 شماره
صفحات -
تاریخ انتشار 2014