The present volume is a translation, revision and updating of our book (pub lished in French)
with the title" Geometrie Algebrique Reelle". Since its pub lication in 1987 the theory has …

For a long time-at least from Fermat to Minkowski-the theory of quadratic forms was a part of
number theory. Much of the best work of the great number theorists of the eighteenth and …

Given that an element a in a ring A is a sum of squares in A, say (l F Xa;, it is natural to ask
for the smallest t (called the length of a in A) for which such an expression is possible. For …

This paper gives a survey, with detailed references to the literature, on recent developments
in real algebra and geometry concerning the polarity between positivity and sums of …

Let $ V $ be an affine algebraic variety over $\mathbb {R} $(or any other real closed field $ R
$). We ask when it is true that every positive semidefinite (psd) polynomial function on $ V …

This textbook originates from a course for graduate students that I taught at Konstanz
University approximately five or six times over the past twenty years. While the first part of the …

Giving an easily accessible elementary introduction to the algebraic theory of quadratic
forms, this book covers both Witt's theory and Pfister's theory of quadratic forms. Leading …

We study a new connection between multidimensional continued fractions, such as Jacobi--
Perron algorithm, and additively indecomposable integers in totally real cubic number fields …

Over a finitely generated field extension in m variables over ap-adic field, any quadratic form
in more than 2 m+ 2 variables has a nontrivial zero. This bound is sharp. We extend this …

The K-length of a form f in K x_ 1, ..., x_ n, K ⊂ C, is the smallest number of d-th powers of
linear forms of which f is a K-linear combination. We present many results, old and new …