Let C be a real plane algebraic curve defined by the resultant of two polynomials (resp. by the discriminant of a polynomial). Geometrically such a curve is the projection of the intersection of the surfaces P (x, y, z) = Q(x, y, z) = 0 (resp. P (x, y, z) = ∂P ∂z (x, y, z) = 0), and generically its singularities are nodes (resp. nodes and ordinary cusp). State-of-the-art numerical algorithms ca...

The task of determining the approximate greatest common divisor (GCD) of more than two univariate polynomials with inexact coefficients can be formulated as computing for a given Bezout matrix a new Bezout matrix of lower rank whose entries are near the corresponding entries of that input matrix. We present an algorithm based on a version of structured nonlinear total least squares (SNTLS) meth...

For nonsingular varieties, the one-dimensional case is an easy fact in curve theory. The two-dimensional case follows from the work of Reider [16]. In higherdimensional cases, (I) is known for n = 3 [3] and n = 4 [8], and by [1] we know that KX + 1 2 (n2 +n+ 2)D is generated by global sections for all n. Less is known about (II) with one exception: if D is already very ample, then (I) and (II) ...

We announce some new results about systems of residue sets. A residue set R C Z is an arithmetic progression R = {a,a±n,a± 2n,...}. The positive integer n is referred to as the modulus of R. Following Znam [21] we denote this set by a(n). We need several number-theoretic functions. p(ra)-the least prime divisor of a natural number ra, P(m)-the greatest prime divisor of ra, A(m)-the greatest div...

Our work is devoted to ramification theory of n-dimensional schemes with n ≥ 2. We start to develop an approach which is very natural; however, it has not got enough attention, except for [Del] and [Bry83]. The idea is to reduce the situation of an n-dimensional scheme X to a number of 1-dimensional settings, just by restricting to curves C properly crossing the ramification subscheme R ⊂ X . W...

The topological filtration on K ′ 0 of a Severi-Brauer variety is computed if the quotient of its index and exponent is a squarefree number and for each prime p dividing this quotient the p -primary component of the corresponding division algebra is decomposable. This gives in particular a description of Ch for such varieties. Let D be a central simple algebra over a field F and X = SB(D) the S...

Gotzmann proved the persistence for minimal growth for ideals. His theorem is called Gotzmann’s persistence theorem. In this paper, based on the combinatorics on binomial coefficients, a simple combinatorial proof of Gotzmann’s persistence theorem in the special case of monomial ideals is given. Introduction Let K be an arbitrary field, R = K[x1, x2, . . . , xn] the polynomial ring with deg(xi)...

An attack is possible upon all three RSA analogue PKCs based on singular cubic curves given by Koyama. While saying so, Seng et al observed that the scheme become insecure if a linear relation is known between two plaintexts. In this case, attacker has to compute greatest common divisor of two polynomials corresponding to those two plaintexts. However, the computation of greatest common divisor...

Some twenty-five years ago, Iskovskih classified the smooth complex Fano threefolds with Picard number one. Apart from P and the quadric, his list includes 5 families of Fano varieties of index two and 11 families of varieties of index one (for index one threefolds, the cube of the anticanonical divisor takes all even values from 2 to 22, except 20). Recently, the author ([A]) and C. Schuhmann ...