An algorithmic approach to
integral quadratic and bilinear forms
Andrzej Mr´oz
Nicolaus Copernicus University, Toru´n, Poland, e-mail: amroz@mat.umk.pl, andrzej@cimat.mx Quadratic forms, that is, homogeneous polynomials of second degree:
q(x1, . . . , xn) =
X
i≤j
qijxixj,
appear in various contexts. Often they are attached to more “complicated” mathematical objects, in such a way that they reflect important properties of these objects, or various clas-sification problems can be reduced to clasclas-sifications of associated quadratic forms (possibly together with some simple additional data). This happens e.g. in representation theory of finite dimensional algebras, theory of Lie groups and algebras, topology/geometry (knots, manifolds, sheaves, singularities,...), graph theory etc. Of course, quadratic forms are also interesting themselves, for example, they induce nice and challenging Diophantine equations of the shape q(x1, . . . , xn) = d, for d ∈ N, in case q has integral coefficients qij.
During this course (consisting of five 1.5h meetings), we are going to discuss chosen properties of quadratic forms (over Z, Q and R), their roots and isotropic vectors, and to give more or less complete proof of a classification of positive (resp. principal) unit integral quadratic forms by Dynkin (resp. Euclidean) graphs. The (sketches of) the proofs will be mostly elementary, constructive, algorithmic, expressed in terms of edge-bipartite graphs (= bigraphs) associated with integral quadratic forms, and so-called inflation algorithm (together with presentations of computer programs), mainly in the flavour of D. Simson’s works.
We will mention also related topics concerning non-unit or indefinite integral quadratic forms, associated bilinear forms, Coxeter formalism and we formulate some open problems.
The presented material in this shape is not included in handbooks, it is quite new and based on recent papers of D. Simson et al., but closely related to older results of M. Barot, H. J. von H¨ohne, S. A. Ovsienko, J. A. de la Pe˜na and other authors. This series of lectures can be treated as a continuation or a supplement to the last semester course on quadratic forms of J. A. de la Pe˜na. Nevertheless, as it will be self-contained, it is addressed also to people not attending to the latter course.
Course addressed to: all interested in the following areas: computational/applied linear algebra, mathematical computing, discrete mathematics, algorithms, graph theory, number theory, computer algebra, representation theory of algebras. The lectures will be in English. Prerequisites: elementary linear algebra (the basics of: matrix calculus, vector spaces and their bases, linear maps and their matrices, eigenvalues, Jordan form). We will try to include some examples and motivations arising from representation theory of algebras, but the knowledge from this area is not indispensable to follow the main ideas of the material.
