# Nonnegative Morse polynomial functions and polynomial optimization

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## Abstract

In this paper we study the representation of Morse polynomial functions which are nonnegative on a compact basic closed semi-algebraic set in $$\mathbb R^{n}$$, and having only finitely many zeros in this set. Following Bivià-Ausina (Math Z 257:745–767, 2007), we introduce two classes of non-degenerate polynomials for which the algebraic sets defined by them are compact. As a consequence, we study the representation of nonnegative Morse polynomials on these kinds of non-degenerate algebraic sets. Moreover, we apply these results to study the polynomial optimization problem for Morse polynomial functions.

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## Notes

1. The polynomial function $$f: \mathbb R^n \rightarrow \mathbb R$$ is said to be convenient if its Newton polyhedron at infinity $$\widetilde{\Gamma }(f)$$ intersects each coordinate axis in a point different from the origin, that is, if for any $$i\in \{1,\ldots ,n\}$$ there exists some integer $$m_i>0$$ such that $$m_i\mathbf {e}_i \in \widetilde{\Gamma }(f)$$. Here $$\{\mathbf {e}_1,\ldots ,\mathbf {e}_n\}$$ denotes the canonical basis in $$\mathbb R^n$$.

2. f is called (Khovanskii) non-degenerate at infinity if for any face $$\Delta$$ of $$\widetilde{\Gamma }(f)$$ which does not contain the origin $$0\in \mathbb R^n$$, the system of equations

\begin{aligned} f_\Delta =x_1\dfrac{\partial f_\Delta }{\partial x_1}=\cdots = x_n\dfrac{\partial f_\Delta }{\partial x_n}=0 \end{aligned}

has no solution in $$(\mathbb R\setminus \{0\})^n$$. Here for $$f=\sum _{\alpha } f_\alpha X^\alpha \in \mathbb R[X]$$, $$f_\Delta :=\sum _{\alpha \in \Delta } f_\alpha X^\alpha$$ denotes the principal part at infinity of f with respect to $$\Delta$$.

## References

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## Acknowledgments

The author would like to express his gratitude to Professor Hà Huy Vui for his valuable discussions on Morse theory and polynomial optimization. The original verson of this work was completed while the author was visiting the Vietnam Institute for Advanced Study in Mathematics (VIASM) in the year 2014 for his postdoctoral fellowship. He thanks VIASM for financial support and hospitality. The work was also supported in part by the grant-aided research project of the Vietnam Ministry of Education and Training. Finally, the author would like to express his warmest thanks to the referees for the careful reading and detailed comments with many helpful suggestions.

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Correspondence to Lê Công-Trình.

Dedicated to Professor Hà Huy Vui on the occasion of his 65th birthday.

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Công-Trình, L. Nonnegative Morse polynomial functions and polynomial optimization. Positivity 20, 483–498 (2016). https://doi.org/10.1007/s11117-015-0367-z

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• DOI: https://doi.org/10.1007/s11117-015-0367-z