A Positivstellensatz for forms on the positive orthant

TL;DR

This paper establishes a Positivstellensatz for forms on the positive orthant, showing that certain powers of forms with positive evaluations have eventually strictly positive coefficients, with proofs based on Handelman's results and real algebra techniques.

Contribution

It proves that powers of forms with positive values on the positive orthant eventually have strictly positive coefficients, extending Positivstellensatz results to this setting.

Findings

For a form p with p(1,...,1)>0, p^m has strictly positive coefficients for large m.

For any form q positive on the positive orthant, p^mq has strictly positive coefficients for large m.

Provides two proofs: one using Handelman's results, another using real algebra techniques.

Abstract

Let $p$ be a nonconstant form in $\mathbb{R}[x_1,\dots,x_n]$ with $p(1,\dots,1)>0$. If $p^m$ has strictly positive coefficients for some integer $m\ge1$, we show that $p^m$ has strictly positive coefficients for all sufficiently large $m$. More generally, for any such $p$, and any form $q$ that is strictly positive on $(\mathbb{R}_+)^n\setminus\{0\}$, we show that the form $p^mq$ has strictly positive coefficients for all sufficiently large $m$. This result can be considered as a strict Positivstellensatz for forms relative to $(\mathbb{R}_+)^n\setminus\{0\}$. We give two proofs, one based on results of Handelman, the other on techniques from real algebra.

Equations

$${\mathbb{R}}_{+}^n:=\{x=(x_1,\dots ,x_n)\in {\mathbb{R}}^n:x_1,\dots ,x_n\ge 0\}.$$

$$E=E_{J,\beta }:=\{w\in S:w_J=\beta \}$$

$$k{F}_J+z=\{w\in {\mathbb{Z}}^n:|w|=ke+|z|,w\ge z,w_J=z_J\},$$

$$A\>=\>\Bigl{\{}\frac{p}{f^{r}}\colon r\geq 0,\ p\in{\mathbb{R}}[\mathsf{x}]_{dr}\Bigr{\}}$$

$${\psi }^m\phi =\frac{l^{md}{x}_1^{k}g}{f^{m+r}}$$

$$p_{\lambda }=(x_1+x_2{)}^{2k}-\lambda x_1^k{x}_2^k+\sum _{\stackrel{|w|=2k}{w_i\ne 0\text{ for some }i\ge 3}}{x}^w\in \mathbb{R}[x_1,\dots ,x_n]$$

Full text

1. Introduction

Given polynomials $p,q\in{\mathbb{R}}[\mathsf{x}]:={\mathbb{R}}[x_{1},\dots,x_{n}]$ where all the coefficients of $p$ are nonnegative, Handelman [8] gave a necessary and sufficient condition (reproduced as Theorem 2.2 below) for there to exist a nonnegative integer $m$ such that the coefficients of $p^{m}q$ are all nonnegative. In another paper [9], Handelman showed that, given a polynomial $p\in{\mathbb{R}}[\mathsf{x}]$ such that $p(1,\dots,1)>0$, if the coefficients of $p^{m}$ are all nonnegative for some $m>0$, then the coefficients of $p^{m}$ are all nonnegative for every sufficiently large $m$.

In the case where $p$ is a form (i.e. a homogeneous polynomial), there is a stronger positivity condition that $p$ may satisfy. If $p(x)=\sum_{|w|=d}a_{w}x^{w}\in{\mathbb{R}}[\mathsf{x}]$ is homogeneous of degree $d$ (with $a_{w}\in{\mathbb{R}}$), we say that $p$ has strictly positive coefficients if $a_{w}>0$ for all $|w|=d$. Here we use standard multi-index notation, where an $n$-tuple $w=(w_{1},\ldots,w_{n})$ of nonnegative integers has length $|w|:=w_{1}+\cdots+w_{n}$ and $x^{w}=x_{1}^{w_{1}}x_{2}^{w_{2}}\cdots x_{n}^{w_{n}}$. Denote the closed positive orthant of real $n$-space by

[TABLE]

Our main result is as follows:

Theorem 1.1**.**

Let $p\in{\mathbb{R}}[\mathsf{x}]$ be a nonconstant real form. The following are equivalent:

1. (A)

The form $p^{m}$ has strictly positive coefficients for some odd $m\geq 1$. 2. (B)

The form $p^{m}$ has strictly positive coefficients for some $m\geq 1$, and $p(x)>0$ at some point $x\in{\mathbb{R}}_{+}^{n}$. 3. (C)

For each real form $q\in{\mathbb{R}}[\mathsf{x}]$ strictly positive on ${\mathbb{R}}_{+}^{n}\setminus\{0\}$, there exists a positive integer $m_{0}$ such that $p^{m}q$ has strictly positive coefficients for all $m\geq m_{0}$.

Theorem 1.1 can be derived from an isometric imbedding theorem for holomorphic bundles, due to Catlin-D’Angelo [5]. The argument is sketched in an appendix at the end of this paper. Another condition equivalent to each of the three conditions Theorem 1.1 was given by the second author and To in [19]. The line of argumentation in [19] is analytic in nature, and the proof therein invokes Catlin-D’Angelo’s isometric embedding theorem.

As the statement of Theorem 1.1 involves only real polynomials, it is desirable to give a purely algebraic proof, which is what we shall do below. We will in fact give two proofs of very different nature. Both are independent of Catlin-D’Angelo’s proof in [5], which uses compactness of the von Neumann operator on pseudoconvex domains of finite-type domains in ${\mathbb{C}}^{n}$ and an asymptotic expansion of the Bergman kernel function by Catlin [4]. We remark that in the case when $n=2$, Theorem 1.1 follows from De Angelis’ work in [6] and has been independently observed by Handelman [10].

Our first proof of Theorem 1.1 uses the criterion of Handelman [8] mentioned above. Our second proof reduces Theorem 1.1 to the archimedean local-global principle due to the first author, in a spirit similar to [16].

For a real form, having strictly positive coefficients is a certificate for being strictly positive on ${\mathbb{R}}_{+}^{n}\setminus\{0\}$. Therefore Theorem 1.1 can be seen as a Positivstellensatz for forms $q$, relative to ${\mathbb{R}}_{+}^{n}\setminus\{0\}$. In particular, the case where $p=x_{1}+\cdots+x_{n}$ specializes to the classical Pólya Positivstellensatz [14] (reproduced in [11, pp. 57–60]). For any $n\geq 2$ and even $d\geq 4$, there are examples of degree-$d$ $n$-ary real forms $p$ with some negative coefficient that satisfy the equivalent conditions of Theorem 1.1 (see Example 5.3 below).

Acknowledgements.

The second author would like to thank his PhD supervisor Professor Wing-Keung To for his continued guidance and support. We would also like to thank David Handelman for his answer on MathOverflow [10], and the anonymous referee for pointing out reference [12].

2. A theorem of Handelman

Let $p=\sum_{w\in{\mathbb{Z}}^{n}}c_{w}x^{w}\in{\mathbb{R}}[\mathsf{x},\mathsf{x}^{-1}]:={\mathbb{R}}[x_{1},\ldots,x_{n},x_{1}^{-1},\ldots,x_{n}^{-1}]$ be a Laurent polynomial. Following Handelman [8] we introduce the following terminology. The Newton diagram of $p$ is the set $\operatorname{Log}(p):=\{w\in{\mathbb{Z}}^{n}:c_{w}\neq 0\}$. A subset $F$ of $\operatorname{Log}(p)$ is a relative face of $\operatorname{Log}(p)$ if there exists a face $K$ of the convex hull of $\operatorname{Log}(p)$ in ${\mathbb{R}}^{n}$ such that $F=K\cap\operatorname{Log}(p)$. In particular, the subset $\operatorname{Log}(p)$ is itself a relative face of $\operatorname{Log}(p)$, called the improper relative face. Given a set $F\subseteq{\mathbb{Z}}^{n}$, an integer $k\geq 1$ and a point $z\in{\mathbb{Z}}^{n}$, we write $kF+z:=\{w^{(1)}+\cdots+w^{(k)}+z\colon w^{(1)},\dots,w^{(k)}\in F\}\subseteq{\mathbb{Z}}^{n}$. For a subset $E$ of ${\mathbb{Z}}^{n}$ and the above Laurent polynomial $p$ we write $p_{E}:=\sum_{w\in E}c_{w}x^{w}$.

Definition 2.1**.**

Let $p\in{\mathbb{R}}[\mathsf{x},\mathsf{x}^{-1}]$ be a nonzero Laurent polynomial. Given a relative face $F$ of $\operatorname{Log}(p)$ and a finite subset $S$ of ${\mathbb{Z}}^{n}$, a stratum of $S$ with respect to $F$ is a nonempty subset $E\subseteq S$ such that

1. (i)

there exist $k\geq 1$ and $z\in{\mathbb{Z}}^{n}$ such that $E\subseteq kF+z$; and 2. (ii)

whenever $E\subseteq kF+z$ for some $z\in{\mathbb{Z}}^{n}$ and some $k\geq 1$, it follows that $E=(kF+z)\cap S$.

A stratum $E$ of $S$ with respect to $F$ is dominant if, in addition, the following holds:

1. (iii)

If $E\subseteq(k\operatorname{Log}(p)+z)\setminus(kF+z)$ for some $k\geq 1$ and some $z\in{\mathbb{Z}}^{n}$, then $(kF+z)\cap S=\emptyset$.

Theorem 2.2** (Handelman [8, Theorem A]).**

Let $p$ and $q$ be Laurent polynomials in ${\mathbb{R}}[\mathsf{x},\mathsf{x}^{-1}]$, where $p$ has nonnegative coefficients. Then $p^{m}q$ has nonnegative coefficients for some positive integer $m$ if, and only if, both the following conditions hold:

1. (a)

For each dominant stratum $E$ of $\operatorname{Log}(q)$ with respect to the improper relative face $\operatorname{Log}(p)$, the polynomial $q_{E}$ is strictly positive on the interior of ${\mathbb{R}}_{+}^{n}$. 2. (b)

For each proper relative face $F$ of $\operatorname{Log}(p)$, and each dominant stratum $E$ of $\operatorname{Log}(q)$ with respect to $F$, there exists a positive integer $m$ such that $p_{F}^{m}q_{E}$ has nonnegative coefficients.

Here, for a Laurent polynomial $f$, by “$f$ has nonnegative coefficients”, we mean that all coefficients of $f$ are nonnegative. As observed in [8], the product of a suitable monomial with $p_{F}$ (resp. $f_{E}$) is a Laurent polynomial involving fewer than $n$ variables (when $F$ is proper), so that the condition (b) is inductive.

3. First proof of Theorem 1.1

We fix an integer $n\geq 1$ and use the notation $\mathsf{x}=(x_{1},\dots,x_{n})$ and $[n]=\{1,\dots,n\}$. Given $z\in{\mathbb{Z}}^{n}$ and a subset $J$ of $[n]$, let $z_{J}:=(z_{j})_{j\in J}\in{\mathbb{Z}}^{J}$ denote the corresponding truncation of $z$. For a nonnegative integer $d$, we write $({\mathbb{Z}}^{n}_{+})_{d}=\{w\in{\mathbb{Z}}^{n}\colon w_{1}\geq 0,\dots,w_{n}\geq 0$, $w_{1}+\cdots+w_{n}=d\}$.

Lemma 3.1**.**

Let $p\in{\mathbb{R}}[\mathsf{x}]$ be a form of degree $d\geq 1$ with strictly positive coefficients. Let $e\geq 0$, and let $S\subseteq({\mathbb{Z}}^{n}_{+})_{e}$ be a nonempty subset.

• (a)

The relative faces of $\operatorname{Log}(p)$ are the sets $F_{J}:=\{w\in({\mathbb{Z}}^{n}_{+})_{d}\colon w_{J}=0\}$, where $J\subseteq[n]$ is a subset.

• (b)

Let $J\subseteq[n]$. For each stratum $E$ of $S$ with respect to $F_{J}$, there exists $\beta\in{\mathbb{Z}}_{+}^{J}$ satisfying $|\beta|\leq e$ such that

[TABLE]

• (c)

If $S=({\mathbb{Z}}_{+}^{n})_{e}$ and $\emptyset\neq J\subsetneq[n]$, the stratum $E_{J,\beta}$ of $S$ with respect to $F_{J}$ is dominant if and only if $\beta=0$.

In particular, $E=S$ is the only stratum of $S$ with respect to the improper relative face $\operatorname{Log}(p)$ of $\operatorname{Log}(p)$, by (b). Note that this stratum is dominant for trivial reasons.

Proof.

By assumption we have $\operatorname{Log}(p)=({\mathbb{Z}}^{n}_{+})_{d}$. Denote this set by $F$. Assertion (a) is clear. Note that $J=\emptyset$ resp. $J=[n]$ gives $F_{J}=F$ resp. $F_{J}=\emptyset$. To prove (b), fix a subset $J\subseteq[n]$, and let $E\subseteq S$ be a stratum of $S$ with respect to $F_{J}$. So there exist $k\geq 1$ and $z\in{\mathbb{Z}}^{n}$ such that $E=(kF_{J}+z)\cap S$. By the particular shape of $F$ we have

[TABLE]

where $w\geq z$ means $w_{i}\geq z_{i}$ for $i=1,\dots,n$. Therefore $E\subseteq E_{J,\beta}$ with $\beta:=z_{J}$. Note that $E_{J,\beta}$ can be nonempty only when $|\beta|\leq e$. The proof of (b) will be completed if we show that $E_{J,\beta}\subseteq lF_{J}+y$ holds for suitable $l\geq 1$ and $y\in{\mathbb{Z}}^{n}$. To this end it suffices to observe that there exist $l\geq 1$ and $y\in{\mathbb{Z}}^{n}$ such that $ld\geq e$, $|y|=e-ld$, $y_{J}=\beta$ and $y_{i}\leq 0$ for $i\in[n]\setminus J$. These $l$ and $y$ will do the job.

It remains to prove (c), so assume now that $S=({\mathbb{Z}}^{n}_{+})_{e}$. Let $J\subsetneq[n]$ be a proper subset, and let $\beta\in{\mathbb{Z}}_{+}^{J}$ be such that $E_{J,\beta}$ is nonempty (hence a stratum of $S$). First assume $\beta\neq 0$. There exist $k\geq 1$ and $z\in{\mathbb{Z}}^{n}$ such that $0\leq z_{J}\leq\beta$ and $z_{J}\neq\beta$, such that $z_{[n]\setminus J}\leq 0$, and such that $|z|=e-kd$. Then $E_{J,\beta}\subseteq kF+z$ and $E_{J,\beta}\cap(kF_{J}+z)=\emptyset$. But there exists $w\in S$ with $w_{J}=z_{J}$, showing that $(kF_{J}+z)\cap S\neq\emptyset$, whence $E_{J,\beta}$ is not dominant. On the other hand, $E_{J,\beta}$ is easily seen to be dominant when $\beta=0$. ∎

We now give a first proof of Theorem 1.1. The implications (A) $\Rightarrow$ (B) and (C) $\Rightarrow$ (A) are trivial. To prove (B) $\Rightarrow$ (C), it suffices to show the following apparently weaker statement:

Lemma 3.2**.**

Given forms $f,\,g\in{\mathbb{R}}[x_{1},\dots,x_{n}]$, where $f$ is nonconstant with strictly positive coefficients and where $g$ is strictly positive on ${\mathbb{R}}^{n}_{+}\setminus\{0\}$, there exists $l\geq 1$ such that $f^{l}g$ has nonnegative coefficients.

Assuming that Lemma 3.2 has been shown, we can immediately state a stronger version of this lemma. Namely, under the same assumptions it follows that $f^{l}g$ actually has strictly positive coefficients for suitable $l\geq 1$. Indeed, choose a form $g^{\prime}$ with $\deg(g^{\prime})=\deg(g)$ such that $g^{\prime}$ has strictly positive coefficients and the difference $h:=g-g^{\prime}$ is strictly positive on ${\mathbb{R}}_{+}^{n}\setminus\{0\}$, for instance $g^{\prime}=c(x_{1}+\cdots+x_{n})^{\deg(g)}$ with sufficiently small $c>0$. Applying Lemma 3.2 to $(f,h)$ instead of $(f,g)$ gives $l\geq 1$ such that $f^{l}h$ has nonnegative coefficients. Since $f^{l}g^{\prime}$ has strictly positive coefficients, the same is true for $f^{l}g=f^{l}g^{\prime}+f^{l}h$.

Now assume that condition (B) of Theorem 1.1 holds. Then the form $p$ is strictly positive on ${\mathbb{R}}_{+}^{n}\setminus\{0\}$. In order to prove (C), apply the strengthened version of Lemma 3.2 to $(f,g)=(p^{m},\,p^{i}h)$ for $0\leq i\leq m-1$. This gives $l\geq 1$ such that $p^{lm+i}h$ has nonnegative coefficients for all $i\geq 0$, which is (C). So indeed it suffices to prove Lemma 3.2.

Proof of Lemma 3.2.

The case $n=1$ is trivial. Suppose that $n>1$ and the above statement holds in less than $n$ variables. Let $\deg(f)=d\geq 1$ and $\deg(g)=e$. As before, choose a form $g^{\prime}$ with $\deg(g^{\prime})=e$ and with strictly positive coefficients such that $h:=g-g^{\prime}$ is strictly positive on ${\mathbb{R}}^{n}_{+}\setminus\{0\}$. This can be done in such a way that $\operatorname{Log}(h)=({\mathbb{Z}}^{n}_{+})_{e}$, i.e. all coefficients of $h$ are nonzero.

We shall verify that the pair $(f,h)$ satisfies the conditions in Theorem 2.2. Since $f$ has strictly positive coefficients, the only (dominant) stratum of $S=\operatorname{Log}(h)$ with respect to $F=\operatorname{Log}(f)$ is $E=S$, by Lemma 3.1. Thus $h_{E}=h$ is strictly positive on ${\mathbb{R}}_{+}^{n}\setminus\{0\}$, so that condition (a) is satisfied. Next, let $J\subseteq[n]$ be a proper nonempty subset. Using the notation of Lemma 3.1, the only dominant stratum of $S=\operatorname{Log}(h)=({\mathbb{Z}}_{+}^{n})_{e}$ with respect to the proper relative face $F_{J}$ of $F=\operatorname{Log}(f)$ is $E:=E_{J,0}=\{w\in S\colon w_{J}=0\}$, according to Lemma 3.1(c). Without loss of generality we may assume $J=\{r+1,\dots,n\}$ for some $1\leq r<n$, where $J$ has cardinality $n-r$. Then $h_{E}$ is a form in ${\mathbb{R}}[x_{1},\dots,x_{r}]$ that is strictly positive on ${\mathbb{R}}_{+}^{r}\setminus\{0\}$, since $h_{E}(x_{1},\dots,x_{r})=h(x_{1},\dots,x_{r},0,\dots,0)>0$ for all $(x_{1},\dots,x_{r})\in{\mathbb{R}}_{+}^{r}\setminus\{0\}$. Moreover, $f_{F_{J}}$ is a form in ${\mathbb{R}}[x_{1},\dots,x_{r}]$ with strictly positive coefficients. By the inductive hypothesis there exists $m\geq 1$ such that all coefficients of $(f_{F_{J}})^{m}h_{E}$ are nonnegative, which shows that $(f,h)$ satisfies condition (b). Therefore, by Theorem 2.2, there exists $l\geq 1$ such that $f^{l}h$ has nonnegative coefficients. ∎

4. Archimedean local-global principle for semirings

Let $A$ be a (commutative unital) ring, and let $T\subseteq A$ be a subsemiring of $A$, i.e. a subset containing $0,\,1$ and closed under addition and multiplication. Recall that $T$ is said to be archimedean if for any $f\in A$ there exists $n\in{\mathbb{Z}}$ with $n+f\in T$, i.e. if $T+{\mathbb{Z}}=A$. The real spectrum $\mathrm{Sper}(A)$ of $A$ (see e.g. [2] 7.1, [13] 2.4) can be defined as the set of all pairs $\alpha=(\mathfrak{p},\leq)$ where $\mathfrak{p}$ is a prime ideal of $A$ and $\leq$ is an ordering of the residue field of $\mathfrak{p}$. Given a semiring $T\subseteq A$, let $X_{A}(T)\subseteq\mathrm{Sper}(A)$ be the set of all $\alpha\in\mathrm{Sper}(A)$ such that $f\geq_{\alpha}0$ for every $f\in T$. We say that $f\in A$ satisfies $f\geq 0$ (resp. $f>0$) on $X_{A}(T)$ if $f\geq_{\alpha}0$ (resp. $f>_{\alpha}0$) for every $\alpha\in X_{A}(T)$.

We recall the archimedean Positivstellensatz in the following form. In a weaker form, this result was already proved by Krivine [12].

Theorem 4.1** ([18] Corollary 2).**

Let $A$ be a ring, and let $T\subseteq A$ be an archimedean semiring containing $\frac{1}{n}$ for some integer $n>1$. If $f\in A$ satisfies $f>0$ on $X_{A}(T)$, then $f\in T$.

We will need to apply Theorem 4.2 below, which is a local-global principle for archimedean semirings. A slightly weaker version of this result was already proved in [3] Theorem 6.5. We give a new proof which is considerably shorter than the proof in [3].

Theorem 4.2**.**

Let $A$ be a ring, let $T\subseteq A$ be an archimedean semiring containing $\frac{1}{n}$ for some integer $n>1$, and let $f\in A$. Assume that for any maximal ideal $\mathfrak{m}$ of $A$ there exists an element $s\in A\setminus\mathfrak{m}$ such that $s\geq 0$ on $X_{A}(T)$ and $sf\in T$. Then $f\in T$.

Proof.

There exists an integer $k\geq 1$ and elements $s_{1},\dots,s_{k}\in A$ with $\langle s_{1},\dots,s_{k}\rangle=\langle 1\rangle$, and with $s_{i}f\in T$ and $s_{i}\geq 0$ on $X_{A}(T)$ for $i=1,\dots,k$. By [15] Prop. 2.7 there exist $a_{1},\dots,a_{k}\in A$ with $\sum_{i=1}^{k}a_{i}s_{i}=1$ and with $a_{i}>0$ on $X_{A}(T)$ ($i=1,\dots,k$). Since $T$ is archimedean, the last condition implies $a_{i}\in T$, by the Positivstellensatz 4.1. It follows that $f=\sum_{i=1}^{k}a_{i}(s_{i}f)\in T$. ∎

5. Second proof of Theorem 1.1

As in the first proof, it suffices to prove Lemma 3.2. So let $f\in{\mathbb{R}}[\mathsf{x}]={\mathbb{R}}[x_{1},\dots,x_{n}]$ be a form of degree $\deg(f)=d\geq 1$ with strictly positive coefficients, say $f=\sum_{|\alpha|=d}c_{\alpha}x^{\alpha}$. Let $S\subseteq{\mathbb{R}}[\mathsf{x}]$ be the semiring consisting of all polynomials with nonnegative coefficients. We shall work with the ring

[TABLE]

of homogeneous fractions of degree zero, considered as a subring of the field ${\mathbb{R}}(\mathsf{x})$ of rational functions. Let $V\subseteq\mathbb{P}^{n-1}$ be the complement of the projective hypersurface $f=0$. Then $V$ is an affine algebraic variety over ${\mathbb{R}}$, with affine coordinate ring ${\mathbb{R}}[V]=A$. As a ring, $A$ is generated by ${\mathbb{R}}$ and by the fractions $y_{\alpha}=\frac{x^{\alpha}}{g}$ where $|\alpha|=d$. Let $T$ be the subsemiring of $A$ generated by ${\mathbb{R}}_{+}$ and by the $y_{\alpha}$ ($|\alpha|=d$). So the elements of $T$ are precisely the fractions $\frac{p}{f^{r}}$, where $r\geq 0$ and $p\in S$ is homogeneous of degree $dr$.

The semiring $T$ is archimedean, as follows from the identity $\sum_{|\alpha|=d}c_{\alpha}y_{\alpha}=1$ and from $c_{\alpha}>0$ for all $\alpha$ ([1] Lemma 1). First we prove Lemma 3.2 under an extra condition.

Lemma 5.1**.**

Let $f,\,g\in{\mathbb{R}}[x_{1},\dots,x_{n}]$ be forms where $f$ is nonconstant with strictly positive coefficients and $g$ is strictly positive on ${\mathbb{R}}^{n}_{+}\setminus\{0\}$. If $\deg(f)$ divides $\deg(g)$, there exists $m\geq 1$ such that $f^{m}g$ has nonnegative coefficients.

Proof.

Suppose that $r$ is a positive integer such that $\deg(g)=r\deg(f)$. Then the fraction $\frac{g}{f^{r}}$ lies in $A$ and is strictly positive on $X_{A}(T)$, since $g$ is positive on ${\mathbb{R}}^{n}_{+}$. Hence the archimedean Positivstellensatz (Theorem 4.1) gives $\frac{g}{f^{r}}\in T$, and clearing denominators we get the desired conclusion. ∎

Remark 5.2*.*

When $\deg(f)=1$, Lemma 5.1 is in fact Pólya’s Positivstellensatz [14]. In this case, our proof above becomes essentially the same as the proof of [14] given by Berr and Wörmann in [1].

For the general case when $\deg(f)$ does not necessarily divide $\deg(g)$, we need a more refined argument as follows. It is similar to the approach in [16].

Proof of Lemma 3.2.

Fix integers $k\geq 0$, $r\geq 0$ such that $k+e=dr$, and consider the fraction $\varphi:=\frac{x_{1}^{k}g}{f^{r}}\in A$. It suffices to show $\varphi\in T$. Indeed, this means that there are $s\geq 0$ and $p\in S$, homogeneous of degree $ds$, such that $\varphi=\frac{p}{f^{s}}$. We may assume $s\geq r$, then $f^{s-r}x_{1}^{k}g$ has nonnegative coefficients. Clearly this implies that $f^{s-r}g$ has nonnegative coefficients.

We prove $\varphi\in T$ by applying the local-global principle 4.2. So let $\mathfrak{m}$ be a maximal ideal of $A$. Then $\mathfrak{m}$ corresponds to a closed point $z$ of the scheme $V$, and hence of $\mathbb{P}^{n-1}$. There exist real numbers $t_{1},\dots,t_{n}>0$ such that the linear form $l=\sum_{i=1}^{n}t_{i}x_{i}$ does not vanish in $z$. Hence the element $\psi:=\frac{l^{d}}{f}$ of $A$ does not lie in $\mathfrak{m}$. On the other hand, $\psi>0$ on $X_{A}(T)$, since $l$ and $f$ are strictly positive on ${\mathbb{R}}^{n}_{+}\setminus\{0\}$.

By Lemma 5.1, applied to $l$ and $g$, there exists an integer $N\geq 1$ for which $l^{N}g\in S$. Choose an integer $m\geq 1$ so large that $md\geq N$. Then

[TABLE]

lies in $T$. From Theorem 4.2 we therefore deduce $\varphi\in T$, as desired. ∎

We conclude with an example, as promised in the introduction.

Example 5.3*.*

For $n\geq 2$ and even $d=2k\geq 4$, the form

[TABLE]

of degree $d$ satisfies the equivalent conditions of Theorem 1.1 and has a negative coefficient (of the monomial $x_{1}^{k}x_{2}^{k}$) whenever $\binom{2k}{k}<\lambda<2^{2k-1}$. Indeed, it suffices to check the case when $n=2$, in which case the vertification follows similarly as in a result of D’Angelo-Varolin [7, Theorem 3].

Appendix: Proof of Theorem

1.1 from Catlin-D’Angelo’s Theorem

In this appendix, we sketch how the results of Catlin-D’Angelo [5] can be used to deduce Theorem 1.1. As in the first and second proofs of Theorem 1.1, it suffices to prove Lemma 3.2.

Let $\mathsf{z}=(z_{1},\ldots,z_{n})$. Denote by ${\mathbb{C}}[\mathsf{z},\overline{\mathsf{z}}]$ the complex polynomial algebra in the indeterminates $z_{1},\dots,z_{n},\overline{z_{1}},\dots,\overline{z_{n}}$. Equipped with conjugation, ${\mathbb{C}}[\mathsf{z},\overline{\mathsf{z}}]$ has the structure of a commutative complex $\ast$-algebra. A polynomial $P\in{\mathbb{C}}[\mathsf{z},\overline{\mathsf{z}}]$ is said to be Hermitian if $P$ equals its conjugate $\overline{P}$. Equivalently, $P$ is Hermitian if and only if $P(z,\overline{z})$ is real for all $z\in{\mathbb{C}}^{n}$. A Hermitian polynomial $P\in{\mathbb{C}}[\mathsf{z},\overline{\mathsf{z}}]$ is said to be positive on ${\mathbb{C}}^{n}\setminus\{0\}$ if $P(z,\overline{z})>0$ for all $z\in{\mathbb{C}}^{n}\setminus\{0\}$. The bidegree of a monomial $z^{\alpha}\overline{z}^{\beta}=z_{1}^{\alpha_{1}}z_{2}^{\alpha_{2}}\cdots z_{n}^{\alpha_{n}}\overline{z}_{1}^{\beta_{1}}\overline{z}_{2}^{\beta_{2}}\cdots\overline{z}_{n}^{\beta_{n}}\in{\mathbb{C}}[\mathsf{z},\overline{\mathsf{z}}]$ is $(|\alpha|,|\beta|)=(\alpha_{1}+\cdots+\alpha_{n},\beta_{1}+\cdots+\beta_{n})$. A bihomogeneous polynomial is a complex linear combination of monomials of the same bidegree. If a bihomogeneous polynomial $P=\sum_{|\alpha|=d,|\beta|=e}a_{\alpha\beta}z^{\alpha}\overline{z}^{\beta}$ is Hermitian, then $d=e$, i.e. $P$ has bidegree $(d,d)$.

From [5, Definition 2], a Hermitian bihomogeneous polynomial $P$ is said to satisfy the strong global Cauchy-Schwarz (in short, SGCS) inequality if $|P(z,\overline{w})|^{2}<P(z,\overline{z})P(w,\overline{w})$ whenever $z,w\in{\mathbb{C}}^{n}$ are linearly independent.

The following result is a special case of [5, Corollary of Theorem 1] (where the matrix $M$ of bihomogeneous polynomials in [5, Corollary of Theorem 1] has size $1\times 1$).

Theorem 5.4** (Catlin-D’Angelo [5, Corollary of Theorem 1]).**

Let $R\in{\mathbb{C}}[\mathsf{z},\overline{\mathsf{z}}]$ be a nonconstant Hermitian bihomogeneous polynomial such that $R$ is positive on ${\mathbb{C}}^{n}\setminus\{0\}$, the domain $\{z\in{\mathbb{C}}^{n}:R(z,\overline{z})<1\}$ is strongly pseudoconvex, and $R$ satisfies the SGCS inequality. Then for each Hermitian bihomogeneous polynomial $Q\in{\mathbb{C}}[\mathsf{z},\overline{\mathsf{z}}]$ positive on ${\mathbb{C}}^{n}\setminus\{0\}$, there exists $l\geq 1$ and polynomials $h_{1},\ldots,h_{N}\in{\mathbb{C}}[\mathsf{z}]\subseteq{\mathbb{C}}[\mathsf{z},\overline{\mathsf{z}}]$ such that $R^{l}Q=\sum_{k=1}^{N}h_{k}\overline{h_{k}}$.

Proof sketch of Lemma 3.2 from

Theorem 5.4.

Let $f=\sum_{|\alpha|=d}c_{\alpha}x^{\alpha}\in{\mathbb{R}}[\mathsf{x}]$ be a form of degree $d$. Suppose that $f$ is nonconstant with strictly positive coefficients. One verifies that $R:=\sum_{|\alpha|=d}c_{\alpha}z^{\alpha}\overline{z}^{\alpha}$ is a nonconstant Hermitian bihomogeneous polynomial that is positive on ${\mathbb{C}}^{n}\setminus\{0\}$, the domain $\{z\in{\mathbb{C}}^{n}:R(z,\overline{z})<1\}$ is strongly pseudoconvex, and that $R$ satisfies the SGCS inequality.

Now suppose that $g=\sum_{|\beta|=e}b_{\beta}x^{\beta}\in{\mathbb{R}}[\mathsf{x}]$ is a form of degree $e$ which is strictly positive on ${\mathbb{R}}^{n}_{+}\setminus\{0\}$. This implies that $Q:=\sum_{|\beta|=e}b_{\beta}z^{\beta}\overline{z}^{\beta}$ is a Hermitian bihomogenous polynomial that is positive on ${\mathbb{C}}^{n}\setminus\{0\}$. Thus we may apply Theorem 5.4 to obtain $l\geq 1$ such that $R^{l}Q=\sum_{k=1}^{N}h_{k}\overline{h_{k}}$ for some polynomials $h_{1},\ldots,h_{N}\in{\mathbb{C}}[\mathsf{z}]\subseteq{\mathbb{C}}[\mathsf{z},\overline{\mathsf{z}}]$. Hence $R^{l}Q=\sum_{|\alpha|=|\beta|=ld+e}a_{\alpha\beta}z^{\alpha}\overline{z}^{\beta}$ for some positive semidefinite Hermitian matrix $A=(a_{\alpha\beta})_{|\alpha|=|\beta|=ld+e}$. Writing $f^{l}g=\sum_{|\gamma|=ld+e}a_{\gamma}^{\prime}x^{\gamma}$, we see that $A$ is in fact the diagonal matrix $\mathrm{diag}(a_{\gamma}^{\prime})_{|\gamma|=ld+e}$. Since $A$ is positive semidefinite, all the coefficients $a_{\gamma}^{\prime}$ of $f^{l}g$ are nonnegative. This completes the proof of Lemma 3.2. ∎