Real radical initial ideals (Q435959)

In Gröbner bases theory, initial ideals play a fundamental role due to the information they retain about the original ideal. Tropical geometry uses initial ideals to understand the combinatorial structure of an ideal. The present paper deals with initial ideals in the context of real algebraic geometry, where Gröbner bases techniques are not sufficient to characterize varieties.NEWLINENEWLINEThe paper explores the consequences of an ideal \(I\subseteq {\mathbb R}[x_1,\dots,x_n]\) having a real radical initial ideal. The main results of the paper are Theorem 2.3, which relates radical initial ideals to the logarithmic limit set of real varieties and other semialgebraic sets; and Theorem 3.3, which gives conditions on a set of polynomials, so that the preorder they generate is stable.NEWLINENEWLINEThe last section of the paper mentions some of the current algorithms for computing real radicals and gives some conditions under which these computations become more tractable.