Positive Polynomials with Special Structure

In this paper, we establish new sufficient conditions for the polynomial f to be SOS in terms of the Newton polyhedron of f (Theorems 2.6 and 2.12). These new sufficient conditions include results which were proved earlier by Lasserre [13, Theorem 3], Fidalgo and Kovacec [6, Theorem 4.3], Ghasemi and Marshall [7, Theorems 2.1 and 2.3], and Ghasemi and Marshall [8, Theorem 2.3].

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Sufficient conditions for a real polynomial to be a sum of squares of polynomials

Let $f: \mathbb {R}^2 \rightarrow \mathbb {R}$ be a polynomial function in two variables of the form $$ f(x,y) = a_0y^D + a_1(x)y^{D-1} + \cdots + a_D(x),$$ where $D = \deg f$ is the degree of $f$. Assume that $f(x,y) \not = 0$ for all $(x,y) \in \mathbb 

Integrability at infinity of negative powers of polynomials in the plane and its application to convergence of Dirichlet series

Let f(x, y) be a polynomial in two variables of the form 
$$\begin{aligned} f(x,y) = a_0y^D + a_1(x)y^{D-1} + \cdots + a_D(x), \end{aligned}$$

where D is the degree of f. For 
$$r > 0$$

, let 
$$\begin{aligned} G^f(r) = \{(x,y) \in \mathbb {R}^2 : |f(x,y)| \le r\}. \end{aligned}$$

We study the distribution of integer points in 
$$G^f(r)$$

. In assuming that f satisfies the so called weekly degenerate condition for main edges of the complete Newton polygon of f, we show that:

Thi Thao Nguyen

Papers

Sufficient conditions for a real polynomial to be a sum of squares of polynomials

Integrability at infinity of negative powers of polynomials in the plane and its application to convergence of Dirichlet series

Newton polygon and distribution of integer points in sublevel sets