A heuristic algorithm, GCDHEU, is described for polynomial GCD computation over the integers. The algorithm is based on evaluation at a single large integer value (for each variable), integer GCD computation, and a single-point interpolation scheme. Timing comparisons show that this algorithm is very efficient for most univariate problems and it is also the algorithm of choice for many problems...

For the given coprime polynomials over integers, we change their coefficients slightly over integers so that they have a greatest common divisor (GCD) over integers. That is an approximate polynomial GCD over integers. There are only two algorithms known for this problem. One is based on an algorithm for approximate integer GCDs. The other is based on the well-known subresultant mapping and the...

We present an impr{wed variant of the matrix-triangularization subresultant prs method [1] fi~r the computation of a greatest comnum divi~w of two polynomials A and B (of degrees m and n, respectively) along with their polynomial remainder ~quence. It is impr~wed in the sense that we obtain complete theoretical results, independent {}f Van Vleck's theorem [13] (which is not always tnle [2, 6]),...

The identification of the relative position of two real coplanar ellipses can be reduced to the identification of the nature of the singular conics in the pencil they define and, in general, their location with respect to these singular conics in the pencil. This latter problem reduces to find the relative location of the roots of univariate polynomials. Since it is usually desired that all gen...

The behavioral-inheritance relations of [7, 8] can be used to compare the life cycles of objects de ned in terms of Petri nets. They yield partial orders on object life cycles (OLCs). Based on these orders, we de ne concepts such as the greatest common divisor and the least common multiple of a set of OLCs. These concepts have practical relevance: In component-based design, work ow management, ...

The discrete Fourier transform of the greatest common divisor � id[a](m) = m � k=1 gcd(k,m)α m , with αm a primitive m-th root of unity, is a multiplicative function that generalizes both the gcd-sum function and Euler’s totient function. On the one hand it is the Dirichlet convolution of the identity with Ramanujan’s sum, � id[a] = id ∗ c•(a), and on the other hand it can be written as a gener...

In this paper we classify curves of genus two over a perfect field k of characteristic two. We find rational models of curves with a given arithmetic structure for the ramification divisor and we give necessary and sufficient conditions for two models of the same type to be k-isomorphic. As a consequence, we obtain an explicit formula for the number of k-isomorphism classes of curves of genus t...

This article provides a new presentation of Barnett’s theorems giving the degree (resp. coefficients) of the greatest common divisor of several univariate polynomials with coefficients in an integral domain by means of the rank (resp. linear dependencies of the columns) of several Bezout-like matrices. This new presentation uses Bezout or hybrid Bezout matrices instead of polynomials evaluated ...

We present a new characterization of semi-bent and bent quadratic functions on finite fields. First, we determine when a GF (2)-linear combination of Gold functions Tr(x i+1) is semi-bent over GF (2), n odd, by a polynomial GCD computation. By analyzing this GCD condition, we provide simpler characterizations of semi-bent functions. For example, we deduce that all linear combinations of Gold fu...

Let Tn(x) and Un(x) be the Chebyshev’s polynomial of the first kind and second kind of degree n, respectively. For n ≥ 1, U2n−1(x) = 2Tn(x)Un−1(x) and U2n(x) = (−1)An(x)An(−x), whereAn(x) = 2n ∏n i=1(x− cos iθ), θ = 2π/(2n + 1). In this paper, we will study the polynomial An(x). Let An(x) = ∑n m=0 an,mx m. We prove that an,m = (−1)k2m ( l k ) , where k = bn−m 2 c and l = b 2 c. We also complete...