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Here is a counterexample: Take $n=2$ variables $X$ and $Y$. Let $L_1,\dots,L_5$ be linear polynomials such that $$S:=\{ (x, y) \in {\mathbb R}^2 ~|~ L_i(xL_i(x,y) \ge 0\}$$ is a pentagon inscribed in the unit circle. Furthermore set $P:=1-X^2-Y^2$. Assume we could write $P$ as the sum of a globally nonnegative quadratic polynomial $Q$ and nonnegative linear combinations of the $L_i$ and $L_iL_j$. Now $P$ vanishes at the vertices of the pentagon and each $L_i$ is nonnegative at these vertices. Therefore $Q$ vanishes also at the vertices. But being a nonnegative quadratic polynomial, $S$ is a sum of squares of linear polynomials which all have also to vanish at the vertices and therefore are identically zero. This shows that $S$ is the zero polynomial. Now notice that each of the $L_i$ and $L_iL_j$ is strictly positive on at least one of the vertices of the pentagon (at which $P$ vanishes, of course). Since $P$ is a nonnegative linear combination of the $L_i$ and $L_iL_j$, this shows that $P=0$.

If the set $S$ defined by the $L_i$ has non-empty interior, then the convex cone of quadratic polynomials which can be written as a globally nonnegative quadratic polynomial $Q$ and nonnegative linear combinations of the $L_i$ and $L_iL_j$ is closed. In fact, this follows from a much more general result on truncated quadratic modules, see e.g. the book of Marshall cited below (Lemma 4.1.4). This implies that, in the above counterexample, even $P+\varepsilon$ for small $\varepsilon>0$ will fail though this polynomial is strictly positive on $S$.

However, there are a lot theorems going into the direction of what you want. You might want to have a look at the following books...

• Marshall: Positive polynomials and sums of squares
• Prestel: Positive polynomials
• Bochnak, Coste, Roy: Real algebraic geometry
• Basu, Pollack, Roy: Algorithms in real algebraic geometry
• Knebusch, Scheiderer: Einführung in die reelle Algebra
• Andradas, Bröcker, Ruiz: Constructible sets in real geometry

...and the following articles...

Also the so-called "S-procedure" could be of interest for you.

8 added 2 characters in body

Here is a counterexample: Take $n=2$ variables $X$ and $Y$. Let $L_1,\dots,L_5$ be linear polynomials such that $$S:={ S:=\{ (x, y) \in {\mathbb R}^2 ~|~ L_i(x) \ge 0}$$ 0\}$$ is a pentagon inscribed in the unit circle. Furthermore set P:=1-X^2-Y^2. Assume we could write P as the sum of a globally nonnegative quadratic polynomial Q and nonnegative linear combinations of the L_i and L_iL_j. Now P vanishes at the vertices of the pentagon and each L_i is nonnegative at these vertices. Therefore Q vanishes also at the vertices. But being a nonnegative quadratic polynomial, S is a sum of squares of linear polynomials which all have also to vanish at the vertices and therefore are identically zero. This shows that S is the zero polynomial. Now notice that each of the L_i and L_iL_j is strictly positive on at least one of the vertices of the pentagon (at which P vanishes, of course). Since P is a nonnegative linear combination of the L_i and L_iL_j, this shows that P=0. If the set S defined by the L_i has non-empty interior, then the convex cone of quadratic polynomials which can be written as a globally nonnegative quadratic polynomial Q and nonnegative linear combinations of the L_i and L_iL_j is closed. In fact, this follows from a much more general result on truncated quadratic modules, see e.g. the book of Marshall cited below (Lemma 4.1.4). This implies that, in the above counterexample, even P+\varepsilon for small \varepsilon>0 will fail though this polynomial is strictly positive on S. However, there are a lot theorems going into the direction of what you want. You might want to have a look at the following books... • Marshall: Positive polynomials and sums of squares • Prestel: Positive polynomials • Bochnak, Coste, Roy: Real algebraic geometry • Basu, Pollack, Roy: Algorithms in real algebraic geometry • Knebusch, Scheiderer: Einführung in die reelle Algebra • Andradas, Bröcker, Ruiz: Constructible sets in real geometry ...and the following articles... Also the so-called "S-procedure" could be of interest for you. 7 added 1 characters in body Here is a counterexample: Take n=2 variables X and Y. Let L_1,\dots,L_5 be linear polynomials such that S := { S:={ (x, y) \in {\mathbb R}^2 | ~|~ L_i(x) \ge 0 } 0}$$ is a pentagon inscribed in the unit circle. Furthermore set $P:=1-X^2-Y^2$. Assume we could write $P$ as the sum of a globally nonnegative quadratic polynomial $Q$ and nonnegative linear combinations of the $L_i$ and $L_iL_j$. Now $P$ vanishes at the vertices of the pentagon and each $L_i$ is nonnegative at these vertices. Therefore $Q$ vanishes also at the vertices. But being a nonnegative quadratic polynomial, $S$ is a sum of squares of linear polynomials which all have also to vanish at the vertices and therefore are identically zero. This shows that $S$ is the zero polynomial. Now notice that each of the $L_i$ and $L_iL_j$ is strictly positive on at least one of the vertices of the pentagon (at which $P$ vanishes, of course). Since $P$ is a nonnegative linear combination of the $L_i$ and $L_iL_j$, this shows that $P=0$.

If the set $S$ defined by the $L_i$ has non-empty interior, then the convex cone of quadratic polynomials which can be written as a globally nonnegative quadratic polynomial $Q$ and nonnegative linear combinations of the $L_i$ and $L_iL_j$ is closed. In fact, this follows from a much more general result on truncated quadratic modules, see e.g. the book of Marshall cited below (Lemma 4.1.4). This implies that, in the above counterexample, even $P+\varepsilon$ for small $\varepsilon>0$ will fail though this polynomial is strictly positive on $S$.

However, there are a lot theorems going into the direction of what you want. You might want to have a look at the following books...

• Marshall: Positive polynomials and sums of squares
• Prestel: Positive polynomials
• Bochnak, Coste, Roy: Real algebraic geometry
• Basu, Pollack, Roy: Algorithms in real algebraic geometry
• Knebusch, Scheiderer: Einführung in die reelle Algebra
• Andradas, Bröcker, Ruiz: Constructible sets in real geometry

...and the following articles...

Also the so-called "S-procedure" could be of interest for you.