We establish linear lower bounds for the complexity of non-trivial, primitive recursive algorithms from piecewise linear given functions. The main corollary is that logtime algorithms for the greatest common divisor from such givens (such as Stein’s) cannot be matched in efficiency by primitive recursive algorithms from the same given functions. The question is left open for the Euclidean algor...

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As usual Z, Q, R and C denote the ring of integers, the rational field, the real field and the complex field respectively. We also let Z = {1, 2, 3, · · · } and C∗ = C \ {0}. For a ∈ Z and n ∈ Z, by (a, n) we mean th greatest common divisor of a and n, if n is odd then the Jacobi symbol ( a n ) is defined in terms of Legendre symbols (see, e.g. [IR]). For x ∈ R, [x] and {x} stand for the integr...

Intermediate coefficient swell is a well-known difficulty with Buchberger’s algorithm for computing Gröbner bases over the rational numbers. p-Adic and modular methods have been successful in limiting intermediate coefficient growth in other computations, and in particular in the Euclidian algorithm for computing the greatest common divisor (GCD) of polynomials in one variable. In this paper we...

Let j(z) = q−1 + 744 + 196884q + · · · denote the usual elliptic modular function on SL2(Z) (q := e throughout). We shall refer to a complex number τ of the form τ = −b+ √ b2−4ac 2a with a, b, c ∈ Z, gcd(a, b, c) = 1 and b −4ac < 0 as a Heegner point, and we denote its discriminant by the integer dτ := b − 4ac. The values of j at such points are known as singular moduli, and they play a substan...

We describe algorithms for computing the greatest common divisor (GCD) of two univariate polynomials with inexactly-known coeecients. Assuming that an estimate for the GCD degree is available (e.g., using an SVD-based algorithm), we formulate and solve a nonlinear optimization problem in order to determine the coeecients of the \best" GCD. We discuss various issues related to the implementation...

Let $(x) be Euler's totient function. The literature on solving the equation cj)0) = n (see [1, pp. 221-223], [2-5], [6, pp. 50-55, problems B36-B42], [7-11], [12, pp. 228-256], and the references therein) can be viewed as a collection of open problems. For n = 2, we essentially have the problem of factoring the Fermat numbers. Another notorious example is Carmichaels conjecture [3, 7] that if ...