Characterization of Polynomials Whose Large Powers have Fully Positive Coefficients

Abstract

We give a criterion which characterizes a real multi-variate Laurent polynomial with full-dimensional smooth Newton polytope to have the property that all sufficiently large powers of the polynomial have fully positive coefficients. Here a Laurent polynomial is said to have fully positive coefficients if the coefficients of its monomial terms indexed by the lattice points of its Newton polytope are all positive. Our result generalizes an earlier result of Colin Tan and the author, which corresponds to the special case when the Newton polytope of the Laurent polynomial is a translate of a standard simplex. The result also generalizes a result of De Angelis, which corresponds to the special case of univariate polynomials. As an application, we also give a characterization of certain polynomial spectral radius functions of the defining matrix functions of Markov chains.

Appendix: An Example for Theorem 1.1

In this appendix, we give an example of a family of polynomials satisfying the three positivity conditions in Theorem 1.1 so that Theorem 1.1 is applicable to the polynomials, but the earlier result of Colin Tan and the author in [23] does not apply to them. The example is modified from D’Angelo-Varolin [8, Theorem 3] and is given by

$$\begin{aligned} p_{\ell , \lambda _1,\lambda _2,\mu }(x_1, x_2) :&= \prod _{i=1,2} \big [(1+x_i)^{2\ell } - \lambda _i x_i^\ell \big ]-\mu x_1^\ell x_2^\ell \nonumber \\ \quad \text {with } \ell \ge 2&\text { and } \left( {\begin{array}{c}2\ell \\ \ell \end{array}}\right)<\lambda _i< 2^{2\ell - 1},~i=1,2,\nonumber \\ \text {and }&~~0<\mu <\big (\lambda _1-\left( {\begin{array}{c}2\ell \\ \ell \end{array}}\right) \big )\big (\lambda _2-\left( {\begin{array}{c}2\ell \\ \ell \end{array}}\right) \big ). \end{aligned}$$

(6.1)

In particular, a simple numerical example is thus given by

$$\begin{aligned} p_{2,7,7,\frac{1}{2}}=(1+4x_1-x_1^2+4x_1^3+x_1^4)(1+4x_2-x_2^2+4x_2^3+x_2^4)-\dfrac{1}{2}x_1^2x_2^2. \end{aligned}$$

(6.2)

The Newton polytope $\Phi$ of $p_{\ell , \lambda _1,\lambda _2,\mu }$ is the square $[0,2\ell ]^2$ (and not a simplex in ${\mathbb {R}}^2$), and the coefficients of $p_{\ell , \lambda _1,\lambda _2,\mu }$ in $x_1^\ell x_2^j$ and $x_1^j x_2^\ell$, $j\in \{0,1,\ldots ,\ell -1,\ell +1\cdots ,2\ell \}$, are negative. Nonetheless the $\Phi$-homogenization of $p_{\ell , \lambda _1,\lambda _2,\mu }$ satisfies the three positivity conditions in Theorem 1.1 (we will skip the verification which is similar to the calculations in [8]), and thus Theorem 1.1 is applicable to $p_{\ell , \lambda _1,\lambda _2,\mu }$. Since $\Phi$ is not a simplex, it follows that the result in [23] does not apply to $p_{\ell , \lambda _1,\lambda _2,\mu }$. One may be tempted to factorize $p_{\ell , \lambda _1,\lambda _2,\mu }(x_1,x_2)$ into a product of the form $q_1(x_1)\cdot q_2(x_2)$ (where $q_i$ is a polynomial in $x_i$, $i=1,2$), and then try to get the conclusion of Theorem 1.1 for $p_{\ell , \lambda _1,\lambda _2,\mu }$ by applying the result in [23] to each factor $q_i$, $i=1,2$. However, a simple argument by comparing coefficients shows that $p_{\ell , \lambda _1,\lambda _2,\mu }$ does not admit such a factorization, and thus this approach will also not work for $p_{\ell , \lambda _1,\lambda _2,\mu }$.