arXiv:1409.6215v1 [math.AG] 22 Sep 2014 Tropical Effective Primary and Dual Nullstellens¨atze Dima Grigoriev 1 , Vladimir V. Podolskii 2,3 1 CNRS, Math´ ematiques, Universit´ e de Lille, France Dmitry.Grigoryev@math.univ-lille1.fr 2 Steklov Mathematical Institute, Moscow, Russia 3 National Research University Higher School of Economics, Moscow, Russia podolskii@mi.ras.ru Abstract Tropical algebra is an emerging field with a number of applica- tions in various areas of mathematics. In many of these applications appeal to tropical polynomials allows to study properties of mathemat- ical objects such as algebraic varieties and algebraic curves from the computational point of view. This makes it important to study both mathematical and computational aspects of tropical polynomials. In this paper we prove tropical Nullstellensatz and moreover we show effective formulation of this theorem. Nullstellensatz is a next natural step in building algebraic theory of tropical polynomials and effective version is relevant for computational aspects of this field. On our way we establish a simple formulation of min-plus and tropi- cal linear dualities. We also observe a close connection between tropical and min-plus polynomial systems. Contents 1 Introduction 2 2 Preliminaries 7 2.1 Min-plus algebra ......................... 7 3 Results Statement 9 3.1 Tropical and Min-plus Nullstellensatz .............. 9 3.2 Linear Duality .......................... 13 3.3 Tropical vs. Min-plus ....................... 16 1