On a tropical dual Nullstellensatz
Consider \(s\) finite, convex sequences of real numbers, i.e., we have \(a_{j,k}\) for \(j=1,\dots, s\); \(k=0,1,\dots, d_j\), such that \(a_{j,k-1}+a_{j,k+1}\geq 2a_{j,k}\). Put \(N=4\sum d_j\) and \(D=\max d_j\). Suppose that there exists a sequence of real numbers \(y_k\) for \(-N\leq k\leq N+D\) such that for all \(|i|\leq N\) and all \(j\), the minimum \(\min_k (y_{k+i}+a_{j,k})\) is attained for at least two different values of \(k\in\{0,1,\dots, d_j\}\). The paper under review proves that then \((y_k)\) can be chosen to be an arithmetic progression.NEWLINENEWLINEIt is explained why it is appropriate to call this a tropical effective dual Nullstellensatz for tropical univariate polynomials. The sequences \((a_{j,k})\) can be thought of as a system of \(s\) tropical polynomials. The existence of \((y_k)\) can be thought of as the solvability of the tropical linear system given by the Cayley matrix of that tropical polynomial system. The existence of a suitable arithmetic progression \((y_k)\) can be thought of as solvability of the polynomial system itself.NEWLINENEWLINEIt is pointed out that the result for \(s=2\) follows from the approach of [\textit{L. F. Tabera}, Rev. Mat. Iberoam. 24, No. 3, 941--961 (2008; Zbl 1216.14048)], which relies on a theorem due to Kapranov, but that approach does not work for general \(s\).NEWLINENEWLINEA multivariate generalization is conjectured. It is pointed out that the direct tropical analog of the Nullstellensatz fails already in the univariate case and therefore it is natural to study the dual version.
scientific article; zbMATH DE number 6019338
scientific article