Sum of squares and the Łojasiewicz exponent at infinity (Q2925542)

Let \(V=V(h_{1},\ldots ,h_{r})\subset \mathbb{R}^{n},\) \(n\geq 2,\) be an unbounded algebraic set and \(f:\mathbb{R}^{n}\rightarrow \mathbb{R}\) be a polynomial. The authors give a constructive proof of the fact that if \(f\) is a positive on \(V\) then \(f|_{V}\) extends to a positive polynomial on the ambient apace \(\mathbb{R}^{n}.\) Precisely, there exists a polynomial \(h\) of the form \(h(x)=\sum_{i=1}^{r}h_{i}(x)^{2}\sigma _{i}(x),\) where \(\sigma _{i}\) are sums of squares of polynomials of degree at most \(p,\) such that \( f(x)+h(x)>0\) for \(x\in \mathbb{R}^{n}.\) They estimate the exponent \(p\) in terms of degree of \(f,\) the degrees of \(h_{i}\) and the Łojasiewicz exponent at infinity of \(f|_{V}.\) They also prove other versions of the above theorem.