[Capco]

### Sum of Squares over Rationals

#### J. Capco, C. Scheiderer

RISC. Technical report, 2019. [url] [pdf]@

author = {J. Capco and C. Scheiderer},

title = {{Sum of Squares over Rationals}},

language = {english},

abstract = {Recently it has been shown that a multivariate (homogeneous) polynomialwith rational coefficients that can be written as a sum of squares offorms with real coefficients, is not necessarily a sum of squares offorms with rational coefficients. Essentially, only one constructionfor such forms is known, namely taking the $K/\Q$-norm of a sufficientlygeneral form with coefficients in a number field $K$. Whether thisconstruction yields a form with the desired properties depends onGalois-theoretic properties of $K$ that are not yet well understood.We construct new families of examples, and we shed new light on somewell-known open questions.},

year = {2019},

institution = {RISC},

length = {0},

url = {https://www3.risc.jku.at/~jcapco/public_files/ss18/sosq.html}

}

**techreport**{RISC5884,author = {J. Capco and C. Scheiderer},

title = {{Sum of Squares over Rationals}},

language = {english},

abstract = {Recently it has been shown that a multivariate (homogeneous) polynomialwith rational coefficients that can be written as a sum of squares offorms with real coefficients, is not necessarily a sum of squares offorms with rational coefficients. Essentially, only one constructionfor such forms is known, namely taking the $K/\Q$-norm of a sufficientlygeneral form with coefficients in a number field $K$. Whether thisconstruction yields a form with the desired properties depends onGalois-theoretic properties of $K$ that are not yet well understood.We construct new families of examples, and we shed new light on somewell-known open questions.},

year = {2019},

institution = {RISC},

length = {0},

url = {https://www3.risc.jku.at/~jcapco/public_files/ss18/sosq.html}

}