Suppose f(x, y), g(x, y) are two polynomials with complex coefficients. The classical Jacobian Conjecture (due to Keller) asserts the following. Conjecture. (Jacobian Conjecture in dimension two) If the Jacobian of the pair (f, g) is a non-zero constant, then the map (x, y) 7→ (f(x, y), g(x, y)) is invertible. Note that the opposite is clearly true, because the Jacobian of any polynomial map is...

A piecewise algebraic curve is a curve defined by the zero set of a bivariate spline function. Given two bivariate spline spaces Sm Δ and S t n Δ over a domain D with a partition Δ, the Bezout number BN m,r;n,t;Δ is defined as the maximum finite number of the common intersection points of two arbitrary piecewise algebraic curves f x, y 0 and g x, y 0, where f x, y ∈ Sm Δ and g x, y ∈ Sn Δ . In ...

We look at arithmetic progressions on elliptic curves known as Huff curves. By an arithmetic progression on an elliptic curve, we mean that either the x or y-coordinates of a sequence of rational points on the curve form an arithmetic progression. Previous work has found arithmetic progressions on Weierstrass curves, quartic curves, Edwards curves, and genus 2 curves. We find an infinite number...

The following article offers an explanation of the relationship of Jacobi and Gauss sums to Fermât and Artin-Schreier curves which is an analogue of the proof of Stickleberger's theorem. 1. Correspondences. The connection between cubic Fermât curves and cubic Jacobi sums was first observed by Gauss [G], who used it to study such sums. That one can compute the number of points on a Fermât curve ...

Counting rational points on Jacobian varieties of hyperelliptic curves over finite fields is very important for constructing hyperelliptic curve cryptosystems (HCC), but known algorithms for general curves over given large prime fields need very long running times. In this article, we propose an extremely fast point counting algorithm for hyperelliptic curves of type y = x + ax over given large...

introduction:  90 sr/ 90 y source  has  been  used  for  the  intravascular  brachytherapy  to  prevent  coronary  restenosis in the patients who have undergone angioplasty. the aim of this research is to determine the  dose distribution of  90 sr/ 90 y source in a water phantom.  materials  and  methods:  in  the  present  work,  mcnp  code  has  been  applied  to  calculate  the  dose  distri...

We discuss 3D Bézier curves with monotone curvature and torsion, generalizing a 2D class of curves by Y. Mineur et al. © 2006 Elsevier B.V. All rights reserved.

Let K be the function field of a connected regular scheme S of dimension 1, and let f : X → Y be a finite cover of projective smooth and geometrically connected curves over K with g(X) ≥ 2. Suppose that f can be extended to a finite cover X → Y of semi-stable models over S (it is known that this is always possible up to finite separable extension of K). Then there exists a unique minimal such c...

We consider the number of zeros of the integral $I(h) = oint_{Gamma_h} omega$ of real polynomial form $omega$ of degree not greater than $n$ over a family of vanishing cycles on curves $Gamma_h:$ $y^2+3x^2-x^6=h$, where the integral is considered as a function of the parameter $h$. We prove that the number of zeros of $I(h)$, for $0 < h < 2$, is bounded above by $2[frac{n-1}{2}]+1$.

K e y w o r d s P u b l i c k e y cryptography, Elliptic curves, Costly computational problems.