Tropical differential equations (Q335863)

Consider a polynomial \(f(x) = \sum_{i \in I \subset \mathbb Z^m} c_ix^i\) in the algebra \(k[x_1, \ldots, x_m]\), where \(k\) is a field with valuation val. The piecewise-linear equation \(g(w) = \min_i val(c_i) + \langle i, w \rangle\) is the tropicalization of \(f\). Zeros of \(g\) are points where this minimum is achieved at least twice, or is infinite. If \(x \in k\) is a zero of \(f\), then \(w =\mathrm{val}(x) \in \mathbb R^m\) is a zero of \(g\). Thus, if \(g\) has no solutions, then \(f\) also has no solutions.NEWLINENEWLINEThis paper pushes this framework to a system of differential equations (DEs). For DEs of Puiseaux series, the valuation map creates a system of tropical DEs. In particular, a tropical differential equation in \(n\) variables of order \(m\) have the form NEWLINE\[NEWLINE(S_1, \ldots, S_n) \mapsto\min_{i=1,\ldots n; j=1,\ldots,m} \left\{a_i^{(j)} + \mathrm{val}_{S_i}(j),a\right\}.NEWLINE\]NEWLINE Here, each \(S_i\) is a set of positive integers, and \(\mathrm{val}_{S_i}(j)\) is the \(j\)-th derivative of \(S_i\), which is the smallest element in \(S_i-j\). A solution to this equation is a tuple \((S_1, \ldots,S_n)\), such that the minimum in the right hand side is attained at least twice, or is infinite. If the tropical system has no solution, then the original system also has no solution.NEWLINENEWLINEThe author shows that the greedy algorithm produces the minimal solution of a tropical DE. He then shows that for systems in one variable, this algorithm is polynomial time. The author generalizes the framework to nonlinear tropical DEs, and show that even in one variable the problem is NP-hard.