In this paper we present a new family of minimal spectrally arbitrary patterns which allow for arbitrary spectrum by using the Nilpotent-Jacobian method introduced in [3]. The novel approach here is that we use the Intermediate Value Theorem to avoid finding an explicit nilpotent realization of the new minimal spectrally arbitrary patterns.

Are every two separated nets in the plane bilipschitz equivalent? In the late 1990s, Burago and Kleiner and, independently, McMullen resolved this beautiful question negatively. Both solutions are based on a construction of a density function that is not realizable as the Jacobian determinant of a bilipschitz map. McMullen’s construction is simpler than the Burago–Kleiner one, and we provide a ...

We present an efficient endomorphism for the Jacobian of a curve C of genus 2 for divisors having a Non disjoint support. This extends the work of Costello in [12] who calculated explicit formulæ for divisor doubling and addition of divisors with disjoint support in JF(C) using only base field operations. Explicit formulæ is presented for this third case and a different approach for divisor dou...

In the early 1990s, Flynn gave an explicit description of the Jacobian of a genus 2 hyperelliptic curve to perform efficient arithmetic on these objects. In this paper, we give a generalization of Flynn’s work when the ground field has characteristic 2. More precisely, we give an explicit description of both the Jacobian and the Kummer surface. We also give (and explain how we found) explicit f...

We provide explicit formulae for realising the group law in Jacobians of superelliptic curves of genus 3 and C3,4 curves. It is shown that two distinct elements in the Jacobian of a C3,4 curve can be added with 150 multiplications and 2 inversions in the field of definition of the curve, while an element can be doubled with 174 multiplications and 2 inversions. In superelliptic curves, 10 multi...

In this paper, dependent and independent variable transformations are introduced to solve a generalized sinh–Gordon equation by using the binary F-expansion method and the knowledge of elliptic equation and Jacobian elliptic functions. Many different new exact solutions such as double periodic wave and complex wave solutions are obtained. Some previous results are extended. 2013 Elsevier Inc. A...

Let X be a nonsingular complex variety and D a reduced effective divisor in X. In this paper we study the conditions under which the formula cSM (1U ) = c(DerX(− logD)) ∩ [X] is true. We prove that this formula is equivalent to a RiemannRoch type of formula. As a corollary, we show that over a surface, the formula is true if and only if the Milnor number equals the Tjurina number at each singul...

We summarise recent advances in techniques for solving Diophantine problems on hyperelliptic curves; in particular, those for finding the rank of the Jacobian, and the set of rational points on the curve.