Consider the Jacobian of a hyperelliptic genus two curve de ned over a nite eld. Under certain restrictions on the endomorphism ring of the Jacobian, we give an explicit description of all non-degenerate, bilinear, anti-symmetric and Galois-invariant pairings on the Jacobian. From this description it follows that no such pairing can be computed more e ciently than the Weil pairing. To establish...

Frobenius published two proofs of a theorem which characterizes irreducible and fully indecomposable matrices in an algebraic manner. It is shown that the second proof, which depends on the Frobenius-Konig theorem, yields a stronger form of the result than the first. Some curious features in Frobenius’s last paper are examined; these include his criticisms of a result due to D. K&rig and the la...

This chapter studies what happens when we have two interacting Frobenius structures. Specifically, we are interested in when they are “maximally incompatible”, or complementary, and give a definition that makes sense in arbitrary monoidal dagger categories in Section 6.1. We will see that it comes down to the standard notion of mutually unbiased bases from quantum information theory in the cate...

Computing roots of scalar polynomials as the eigenvalues of Frobenius companion matrices using backward stable eigenvalue algorithms is a classical approach. The introduction of new families of companion matrices allows for the use of other matrices in the root-finding problem. In this paper, we analyze the backward stability of polynomial root-finding algorithms via Fiedler companion matrices....

In its simplest form, the coin problem is this: what is the largest positive amount of money that cannot be obtained using two coins of specified distinct denominations? For example, using coins of 2 units and 3 units, it is easy so see that every amount greater than or equal to 2 can be obtained, but 1 cannot be obtained. Using coins of 2 units and 5 units, every amount greater than or equal t...

In this paper, we give a new formulation of invariant theory for elliptic Weyl group using the group O(2, n). As an elliptic Weyl group quotient, we define a suitable C∗-bundle. We show that it has a conformal Frobenius structure which we define in this paper. Then its good section could be identified with a Frobenius manifold which we constructed in [13].

This article addresses some foundational issues that arise in the study of linear codes defined over finite rings. Linear coding theory is particularly well-behaved over finite Frobenius rings. This follows from the fact that the character module of a finite ring is free if and only if the ring is Frobenius.