Definite quadratic form | EPFL Graph Search

In mathematics, a definite quadratic form is a quadratic form over some real vector space V that has the same sign (always positive or always negative) for every non-zero vector of V. According to that sign, the quadratic form is called positive-definite or negative-definite. A semidefinite (or semi-definite) quadratic form is defined in much the same way, except that "always positive" and "always negative" are replaced by "never negative" and "never positive", respectively. In other words, it may take on zero values for some non-zero vectors of V. An indefinite quadratic form takes on both positive and negative values and is called an isotropic quadratic form. More generally, these definitions apply to any vector space over an ordered field. Quadratic forms correspond one-to-one to symmetric bilinear forms over the same space. A symmetric bilinear form is also described as definite, semidefinite, etc. according to its associated quadratic form. A quadratic form Q and its associated symmetric bilinear form B are related by the following equations: The latter formula arises from expanding As an example, let , and consider the quadratic form where and c_1 and c_2 are constants. If and the quadratic form Q is positive-definite, so Q evaluates to a positive number whenever If one of the constants is positive and the other is 0, then Q is positive semidefinite and always evaluates to either 0 or a positive number. If and or vice versa, then Q is indefinite and sometimes evaluates to a positive number and sometimes to a negative number. If and the quadratic form is negative-definite and always evaluates to a negative number whenever And if one of the constants is negative and the other is 0, then Q is negative semidefinite and always evaluates to either 0 or a negative number. In general a quadratic form in two variables will also involve a cross-product term in x_1·x_2: This quadratic form is positive-definite if and negative-definite if and and indefinite if It is positive or negative semidefinite if with the sign of the semidefiniteness coinciding with the sign of This bivariate quadratic form appears in the context of conic sections centered on the origin.

Anneau de Witt des G-formes et multiples de G-formes trace

Philippe Chabloz

This thesis deals with the study of G-forms and particulary the trace form of a G-Galois algebra. Let k be a field of characteristic not two. Let G be a finite group and L a G-Galois algebra over k. We define the trace form qL by qL(x, y) = TrL/k(xy) for all x, y in L. This is a bilinear symmetric form which is G-invariant. In other words, qL is a G-form. We know that L has a self-dual normal basis if and only if the trace form qL is G-isomorphic to the unit G-form q0. This is an important reason to classify the trace forms. This work contains two different parts. In the first part, we study G-forms in general, putting a ring structure on their Witt group. We then proved an analogue of Pfister's theorem - i.e. there is no zero divisor of odd dimension - when k[G] is semi-simple, k is big enough and G is abelian. Counter-examples are given when these conditions are not fulfilled. In the second part of this thesis, we study the trace form qL of a G-Galois algebra. E. Bayer-Fluckiger and H. W. Lenstra proved that if G is of odd order, then qL is always G-isomorphic to the unit form. If G is of even order, this is no longer the case. However, if the field k is of cohomological 2-dimension less than or equal to 1, then E. Bayer-Fluckiger and J.-P. Serre gave a necessary and sufficient condition - in terms of cohomological invariants - for the trace form qL to be isomorphic to the unit form. M. Monsurrò generalized this result to fields of virtual cohomological 2-dimension equal to 1. However, in higher cohomological dimensions, it becomes very difficult to classify the trace form itself. But it is possible to give general results if we consider multiples of the trace form or more generally the product of the trace form by a quadratic form. E. Bayer-Fluckiger formulated 2 conjectures about the possibility of finding a complete system of invariants for such a product when the quadratic form lies in a certain ideal of the Witt ring of k depending on the cohomological dimension of the field. In this work, we prove the first conjecture for all cohomological dimensions and the second one for a field of cohomological 2-dimension equal to 2. A more general conjecture is proved including the fields of virtual cohomological 2-dimension equal to 2. Finally, the second conjecture of E. Bayer-Fluckiger is proved in all cohomological dimensions, but only when either the characteristic of k is non zero or the group G is abelian or a 2-group, or k is big enough.

EPFL2004

Anneau de Witt des G-formes et multiples de G-formes trace

Philippe Chabloz

EPFL2004