In this text, we study factorizations of polynomials over the tropical hyperfield and the sign hyperfield, which we call tropical polynomials and sign polynomials, respectively. We classify all irreducible polynomials in either case. We show that tropical polynomials factor uniquely into irreducible factors, but that unique factorization fails for sign polynomials. We describe division algorithms for tropical and sign polynomials by linear terms that correspond to roots of the polynomials.

中文翻译：

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热带和符号多项式的因式分解

在本文中，我们研究了热带超场和符号超场上多项式的因式分解，我们分别称之为热带多项式和符号多项式。在任何一种情况下，我们都对所有不可约多项式进行分类。我们表明，热带多项式可以唯一地分解为不可约因子，但对于符号多项式来说，唯一的因式分解失败。我们通过与多项式根相对应的线性项来描述热带和符号多项式的除法算法。