## Abstract

In this work, we explore the relation between the tropicalization of a real semi-algebraic set S={f_{1}<0,…,f_{k}<0} defined in the positive orthant and the combinatorial properties of the defining polynomials f_{1},…,f_{k}. We describe a cone that depends only on the face structure of the Newton polytopes of f_{1},…,f_{k} and the signs attained by these polynomials. This cone provides an inner approximation of the real tropicalization, and it coincides with the real tropicalization if S={f<0} and the polynomial f has generic coefficients. Furthermore, we show that for a maximally sparse polynomial f the real tropicalization of S={f<0} is determined by the outer normal cones of the Newton polytope of f and the signs of its coefficients. Our arguments are valid also for signomials, that is, polynomials with real exponents defined in the positive orthant.

Originalsprog | Engelsk |
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Artikelnummer | 107564 |

Tidsskrift | Journal of Pure and Applied Algebra |

Vol/bind | 228 |

Udgave nummer | 6 |

ISSN | 0022-4049 |

DOI | |

Status | Udgivet - 2024 |

### Bibliografisk note

Funding Information:The author thanks Elisenda Feliu for useful discussions and comments on the manuscript. Funded by the European Union under the Grant Agreement no. 101044561 , POSALG. Views and opinions expressed are those of the author(s) only and do not necessarily reflect those of the European Union or European Research Council (ERC). Neither the European Union nor ERC can be held responsible for them

Funding Information:

The author thanks Elisenda Feliu for useful discussions and comments on the manuscript. Funded by the European Union under the Grant Agreement no. 101044561, POSALG. Views and opinions expressed are those of the author(s) only and do not necessarily reflect those of the European Union or European Research Council (ERC). Neither the European Union nor ERC can be held responsible for them

Publisher Copyright:

© 2023 The Author(s)