{
"id": "1910.04147",
"version": "v1",
"published": "2019-10-09T17:47:02.000Z",
"updated": "2019-10-09T17:47:02.000Z",
"title": "On sum of squares certificates of non-negativity on a strip",
"authors": [
"Paula Escorcielo",
"Daniel Perrucci"
],
"categories": [
"math.AG"
],
"abstract": "A well-known result of Murray Marshall states that for every $f \\in \\mathbb{R} [X,Y]$ non-negative on the strip $[0,1] \\times \\mathbb{R}$ can be written as $f= \\sigma_0 + \\sigma_1 X(1-X)$ with $\\sigma_0, \\sigma_1$ sums of squares in $\\mathbb{R} [X,Y]$. In this work, we present a few results concerning this representation in particular cases. First, under the assumption ${\\rm deg}_Y f \\leq 2$, by characterizing the extreme rays of a suitable cone, we obtain a degree bound for each term. Then, we consider the case of $f$ positive on $[0,1] \\times \\mathbb{R}$ and non-vanishing at infinity, and we show again a degree bound for each term, coming from a constructive method to obtain the sum of squares representation. Finally, we show that this constructive method also works in the case of $f$ having only a finite number of zeros, all of them lying on the boundary of the strip, and such that $\\frac{\\partial f}{\\partial X}$ does not vanish at any of them.",
"revisions": [
{
"version": "v1",
"updated": "2019-10-09T17:47:02.000Z"
}
],
"analyses": {
"subjects": [
"12D15",
"13J30",
"14P10"
],
"keywords": [
"squares certificates",
"non-negativity",
"degree bound",
"murray marshall states",
"constructive method"
],
"note": {
"typesetting": "TeX",
"pages": 0,
"language": "en",
"license": "arXiv",
"status": "editable"
}
}
}