We provide a criterion for a coherent sheaf to be an Ulrich sheaf in terms of a certain bilinear form on its global sections. When working over the real numbers, we call it a positive Ulrich sheaf if this bilinear form is symmetric or Hermitian and positive-definite. In that case, our result provides a common theoretical framework for several results in real algebraic geometry concerning the existence of algebraic certificates for certain geometric properties. For instance, it implies Hilbert’s theorem on nonnegative ternary quartics, via the geometry of del Pezzo surfaces, and the solution of the Lax conjecture on plane hyperbolic curves due to Helton and Vinnikov.

A generalization of Schmüdgen’s Positivstellensatz is given which holds for any basic closed semialgebraic set in ${{\mathbb{R}}^{n}}$ (compact or not). The proof is an extension of Wörmann’s proof.

To each field $F$ of characteristic not 2, one can associate a certain Galois group ${{\mathcal{G}}_{F}}$, the so-called $\text{W}$-group of $F$, which carries essentially the same information as the Witt ring $W(F)$ of $F$. In this paper we investigate the connection between ${{\mathcal{G}}_{F}}$ and ${{\mathcal{G}}_{F(\sqrt{a})}}$, where $F(\sqrt{a})$ is a proper quadratic extension of $F$. We obtain a precise description in the case when $F$ is a pythagorean formally real field and $a=-1$, and show that the $\text{W}$-group of a proper field extension $K/F$ is a subgroup of the $\text{W}$-group of $F$ if and only if $F$ is a formally real pythagorean field and $K=F(\sqrt{-1)}$. This theorem can be viewed as an analogue of the classical Artin-Schreier’s theorem describing fields fixed by finite subgroups of absolute Galois groups. We also obtain precise results in the case when $a$ is a double-rigid element in $F$. Some of these results carry over to the general setting.

A preordering $T$ is constructed in the polynomial ring $A=\mathbb{R}[{{t}_{1}},{{t}_{2}},...]$ (countablymany variables) with the following two properties: (1) For each $f\,\in \,A$ there exists an integer $N$ such that $-\,N\,\le \,f\left( p \right)\,\le \,N$ holds for all $P\in \text{Spe}{{\text{r}}_{T}}(A)$. (2) For all $f\,\in \,A$, if $N+f,N-f\in T$ for some integer $N$, then $f\,\in \,\mathbb{R}$. This is in sharp contrast with the Schmüdgen-Wörmann result that for any preordering $T$ in a finitely generated $\mathbb{R}$-algebra $A$, if property (1) holds, then for any $f\in A,f>0\,\text{on}\,\text{Spe}{{\text{r}}_{T}}(A)\Rightarrow f\in T$. Also, adjoining to $A$ the square roots of the generators of $T$ yields a larger ring $C$ with these same two properties but with $\sum{{{C}^{2}}}$ (the set of sums of squares) as the preordering.