Václav Šimerka: quadratic forms and factorization

A New Factorization Technique Using Quadratic Forms

The paper presents a practical method for factoring an arbitrary N by representing N or \N by one of at most three quadratic forms: XAf = s2 — Dy2, X = 1, — 1, 2, D = — 1, ±2, ±3, ±6. These three forms appropriate to N, together with inequalities for y, are given for all N prime to 6. Presently available sieving facilities make the method quite effective and economical for numbers N having 20 t...

Applications of quadratic D-forms to generalized quadratic forms

In this paper, we study generalized quadratic forms over a division algebra with involution of the first kind in characteristic two. For this, we associate to every generalized quadratic from a quadratic form on its underlying vector space. It is shown that this form determines the isotropy behavior and the isometry class of generalized quadratic forms.

A Probabilistic Factorization Algorithm with Quadratic Forms of Negative Discriminant

We propose a probabilistic algorithm for factorization of an integer N with run time (exp^log/V loglogJV)/5/4"1""'1'. Asymptotically, our algorithm will be as fast as the wellknown factorization algorithm of Morrison and Brillhart. The latter algorithm will fail in several cases and heuristic assumptions are needed for its run time analysis. Our new algorithm will be analyzed under the assumpti...

Quadratic nonnegative matrix factorization

In Nonnegative Matrix Factorization (NMF), a nonnegative matrix is approximated by a product of lower-rank factorizing matrices. Most NMF methods assume that each factorizing matrix appears only once in the approximation, thus the approximation is linear in the factorizing matrices. We present a new class of approximative NMF methods, called Quadratic Nonnegative Matrix Factorization (QNMF), wh...

Quadratic Functions, Optimization, and Quadratic Forms