We define the concept of Tschirnhaus-Weierstrass curve, named after the Weierstrass form of an elliptic curve and Tschirnhaus transformations. Every pointed curve has a Tschirnhaus-Weierstrass form, and this representation is unique up to a scaling of variables. This is useful for computing isomorphisms between curves.

We analyze cohomological properties of the Krichever map and use the results to study Weierstrass cycles in moduli spaces and the tautological ring. Let us consider a point p on a smooth projective connected curve C over C of genus g. We say that a natural number n is a non-gap if there exists a function that is holomorphic on C \p and has a pole of order n at the point p (in other words h0(O(n...

The question how the classical definition of the Smith zeros of an LTI continuous-time singular control system S(E, A, B, C, D) can be generalized and related to state-space methods is discussed. The zeros are defined as those complex numbers for which there exists a zero direction with a nonzero state-zero direction. Such a definition allows an infinite number of zeros (then the system is call...

In this note, we investigate the relationship between the finite and infinite frequency structure of a regular polynomial matrix and that of a particular linearization, called the generalised companion matrix. A special resolvent decomposition of the regular polynomial matrix is proposed which is based on the Weierstrass canonical form of this generalised companion matrix and the solution of a ...

On plane algebraic curves the so-called Weierstrass kernel plays the same role of the Cauchy kernel on the complex plane. A straightforward prescription to construct the Weierstrass kernel is known since one century. How can it be extended to the case of more general curves obtained from the intersection of hypersurfaces in a n dimensional complex space? This problem is solved in this work in t...

In this note, we investigate the relationship between the finite and infinite frequency structure of a regular polynomial matrix and that of a particular linearization, called the generalised companion matrix. A special resolvent decomposition of the regular polynomial matrix is proposed which is based on the Weierstrass canonical form of this generalised companion matrix and the solution of a ...

The famous primary and cyclic decomposition theorems along with the tightly related rational and Jordan canonical forms are extended to linear spaces of infinite dimensions with counterexamples showing the scope of extensions.

A “Composition map” is constructed, leaning heavily on earlier work Fittouhi and Joseph (2023) [5], [6]). It defines a composition tableau which recovers the “canonical” Weierstrass section e+V described in first paper above. Moreover without reference to this work, it then shown that indeed section. This results huge simplification. one may read off from “VS quadruplets” of second above papers...

Elliptic curves in Hesse form admit more suitable arithmetic than ones in Weierstrass form. But elliptic curve cryptosystems usually use Weierstrass form. It is known that both those forms are birationally equivalent. Birational equivalence is relatively hard to compute. We prove that elliptic curves in Hesse form and in Weierstrass form are linearly equivalent over initial field or its small e...

A bstract We find closed-form expressions for the Schur indices of 4d $$ \mathcal{N} N = 2 * super Yang-Mills theory with unitary gauge groups arbitrary ranks via Fermi-gas formulation. They can be written as a sum over Young diagrams associated spectral zeta functions an ideal system. These are expressed in ...