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Let $p \in \mathbb{R}[x_1, \ldots, x_n]$ be irreducible. If there is a point $v \in \mathbb{R}^n$ such that $p(v)=0$ and sucht that the gradient $\nabla p(v) \neq 0$. Then, by the Artin-Lang-Theorem, …

Let $n \geq 2$. Let $g_1, \ldots , g_{n-1} \in \mathbb{R}[x_1,\ldots,x_n]$ such that $q=g_1^2+\ldots +g_{n-1}^2$ is not divisible by $p=x_1^2+\ldots +x_n^2$. Let $m \geq 1$ be the smallest integer suc …

Let $V \subseteq \mathbb{R}[x,y]_d$ be a two dimensional linear subspace of the vector space of bivariate forms of degree $d$. For each degree $d$ we can find such subspaces with the property that eve …

It is classically known that every positive integer is a sum of at most four squares of integers, i.e. every sum of squares of integers is a sum of four squares of integers. Now consider a symmetric $ …

Let $p \in \mathbb{R}[x_1, \ldots, x_n]$. The set $S=\{x \in \mathbb{R}^n: p(x) \geq 0\}$ is compact if and only if there is a natural number $N$ and polynomials $g_i, h_i \in \mathbb{R}[x_1, \ldots, …