---
abstract: 'Given two positive definite forms $f,\,g\in{{\mathbb{R}}}[x_1,\dots,x_n]$, we prove that $fg^r$ lies in the interior of the sums of squares cone for large $r$.'
address: 'Fachbereich Mathematik und Statistik, Universität Konstanz, Germany'
author:
- Claus Scheiderer
title: An observation on positive definite forms
---

Fix $n\ge1$ and write ${{\mathbb{R}}}[{{\mathtt{x}}}]={{\mathbb{R}}}[x_1,\dots,x_n]$. A form (homogeneous polynomial) $f\in{{\mathbb{R}}}[{{\mathtt{x}}}]$ is *positive definite* if $f(\xi)>0$ for every $0\ne\xi\in{{\mathbb{R}}}^n$. For every integer $d\ge0$ let ${{\mathbb{R}}}[{{\mathtt{x}}}]_d$ denote the space of forms of degree $d$. Let $\Sigma_{2d}{\subseteq}{{\mathbb{R}}}[{{\mathtt{x}}}]_{2d}$ be the set of all forms of degree $2d$ that are *sos*, i.e. sums of squares of forms. It is well known that $\Sigma_{2d}$ is a full-dimensional closed convex cone in ${{\mathbb{R}}}[{{\mathtt{x}}}]_{2d}$.

The main result of [@sch:ppss] implies (see [@sch:ppss] Remark 4.6):

\[ppssthm\]Let $f,\,g$ be positive definite forms in ${{\mathbb{R}}}[x_1,\dots,x_n]$, with $g$ not constant. Then there is $r_0\ge0$ such that the form $fg^r$ is a sum of squares for all $r\ge r_0$.

A form $f$ of degree $2d$ will be said to be a *strict sum of squares*, or *strictly sos*, if $f$ lies in the interior of the cone $\Sigma_{2d}$. It is equivalent that $f$ has a sum of squares representation $f=f_1^2+\cdots+f_N^2$ in which $f_1,\dots,f_N$ form a linear basis of ${{\mathbb{R}}}[{{\mathtt{x}}}]_d$. The purpose of this note is to observe that Theorem \[ppssthm\] can be sharpened as follows:

\[result\]Let $f,\,g$ be positive definite forms in ${{\mathbb{R}}}[x_1,\dots,x_n]$, with $g$ not constant. Then there is $r_0\ge0$ such that the form $fg^r$ is a strict sum of squares for all $r\ge r_0$.

I am grateful to Amirali Ahmadi for valuable remarks and for suggesting this question.

Before giving the proof we need a bit of preparation. Given a form $f\in\Sigma_{2d}$, let $U_f$ be the set of forms $p\in{{\mathbb{R}}}[{{\mathtt{x}}}]_d$ for which there exists a real number $c>0$ such that $f-cp^2$ is a sum of squares. Then $U_f$ is a linear subspace of ${{\mathbb{R}}}[{{\mathtt{x}}}]_d$, and $f$ is strictly sos if and only if $U_f={{\mathbb{R}}}[{{\mathtt{x}}}]_d$. More generally, the faces of the cone $\Sigma_{2d}$ are precisely the sets $F_U\Sigma_{2d}:=
\{f\in\Sigma_{2d}\colon U_f{\subseteq}U\}$, for $U$ a subspace of ${{\mathbb{R}}}[{{\mathtt{x}}}]_d$. (We will not use this fact.) If $U,\,V$ are linear subspaces of ${{\mathbb{R}}}[{{\mathtt{x}}}]$, let $UV$ denote the linear subspace spanned by the products $uv$ with $u\in U$ and $v\in V$. Let $f$ and $g$ be two forms that are sums of squares. Clearly we have $U_fU_g{\subseteq}U_{fg}$, and $U_f+U_g{\subseteq}U_{f+g}$ if $\deg(f)=\deg(g)$. In particular, the product of two forms that are strictly sos is again strictly sos.

By Theorem \[ppssthm\] there is $k\ge0$ such that $fg^k$ and $fg^{k+1}$ are sos. Replacing $f$ with $fg^k$ or $fg^{k+1}$ and $g$ with $g^2$, we can assume that both forms $f$ and $g$ are sos.

Let $\deg(f)=2d$ and $\deg(g)=2e$, write $A={{\mathbb{R}}}[{{\mathtt{x}}}]$. Let $q$ be a strictly sos form with $\deg(q)=2e$ such that $g-q$ is positive definite, for example $q=c(x_1^2+\cdots+x_n^2)^e$ for suitable real $c>0$. By Theorem \[ppssthm\] there exists $k\ge1$ such that the forms $g^k(g-q)$ and $q^k(g-q)$ are both sos. Then for every $r\ge2k$ the form $$g^r-q^r\>=\>\sum_{j=0}^{r-1}(g-q)g^jq^{r-1-j}$$ is sos, since this is true for every summand on the right. Since $q^r$ is strictly sos, this implies that $g^r$ is strictly sos for every $r\ge2k$.

Fix $p\in A_d$. There is a real number $c>0$ for which $f-cp^2$ is positive definite. By Theorem \[ppssthm\] there is an integer $r(p)\ge2k$ such that $(f-cp^2)g^r$ is sos for $r\ge r(p)$. This implies $U_{p^2}U_{g^r}{\subseteq}U_{fg^r}$ for these $r$, and therefore $pA_{re}{\subseteq}U_{fg^r}$ since $g^r$ is strictly sos. Repeat this argument for every monomial $p$ of degree $d$, and let $r_0$ be the maximum of the respective numbers $r(p)$. For every $r\ge r_0$ we then have $A_dA_{re}=A_{d+re}{\subseteq}U_{fg^r}$, which means that $fg^r$ is strictly sos.

Stengle’s Positivstellensatz says that a polynomial $f$ has strictly positive values if and only if there exist sums of squares of polynomials $g$ and $h$ such that $fg=1+h$. For a form, being a strict sum of squares is a certificate for being positive definite. Therefore Proposition \[result\] can be seen as a strong homogeneous version of Stengle’s Positivstellensatz, in which the multiplier can be chosen to be a power of any preassigned (nonconstant) positive definite form.

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C. Scheiderer: A Positivstellensatz for projective real varieties. Manuscr. math. **138**, 73–88 (2012).
