Let M be a quadruply-punctured sphere with boundary components A; B; C; D. The SL(2; C)-character variety of M consists of equivalence classes of homomorphisms of 1 (M) ?! SL(2; C) and can be identiied with a quartic hypersurface in C 7. corresponding to representations with tr((A)) = a, tr((B)) = b, tr((C)) = c, tr((D)) = d is a cubic surface in C 3. We determine the singular

In this paper we present a method of circular arc approximation by quartic Bézier curve. Our quartic approximation method has a smaller error than previous quartic approximation methods due to the alternation of the error function of our quartic approximation. Our method yields a closed form of error so that subdivision algorithm is available, and curvaturecontinuous quartic spline under the su...

The simplest quartic field was introduced by M. Gras and studied by A. J. Lazarus. In this paper, we will evaluate the values of the Dedekind zeta functions at s = −1 of the simplest quartic fields. We first introduce Siegel’s formula for the values of the Dedekind zeta function of a totally real number field at negative odd integers, and will apply Siegel’s formula to the simplest quartic fiel...

Abstract Presented is a very detailed two-tier analysis of the location real roots general quartic equation $$x^4 + x^3 b x^2 c x d = 0$$ x 4 + a 3 b 2 c d</mml:...

The Q-factoriality of a nodal quartic 3-fold implies its non-rationality. We prove that a nodal quartic 3-fold with at most 8 nodes is Q-factorial, and we show that a nodal quartic 3-fold with 9 nodes is not Q-factorial if and only if it contains a plane. However, there are non-rational non-Q-factorial nodal quartic 3-folds in P. In particular, we prove the non-rationality of a general non-Q-fa...

This paper proposes explicit formulae for the addition step and doubling step in Miller’s algorithm to compute Tate pairing on Jacobi quartic curves. We present a geometric interpretation of the group law on Jacobi quartic curves, which leads to formulae for Miller’s algorithm. The doubling step formula is competitive with that for Weierstrass curves and Edwards curves. Moreover, by carefully c...

The moduli space M0 of semi-stable rank 2 vector bundles with fixed trivial determinant over a non-hyperelliptic curve C of genus 3 is isomorphic to a quartic hypersurface in P7 (Coble’s quartic). We show that M0 is self-dual and that its polar map associates to a stable bundle E ∈ M0 a bundle F which is characterized by dimH 0(C,E ⊗F ) = 4. The projective space PH0(C,E ⊗ F ) is equipped with a...

In this paper we present an approximation method of quadric surface using quartic spline. Our method is based on the approximation of quadratic rational Bézier patch using quartic Bézier patch. We show that our approximation method yields G (tangent plane) continuous quartic spline surface. We illustrate our results by the approximation of helicoid-like surface.

A new approximation method for conic section by quartic Bézier curves is proposed. This method is based on the quartic Bézier approximation of circular arcs. We give the upper bound of Hausdorff distance between the conic section and the quartic Bézier curve, and also show that the approximation order is eight. And we prove that our approximation method has a smaller upper bound than previous q...