A simple counterexample to Kouchnirenko's conjecture (Q1610956)

The author gives a counter-example to a conjecture of Kouchnirenko on an upper bound for the number of nondegenerate isolated positive roots of a system of \(k\) polynomial equations in \(k\) variables \(f_1 = \cdots = f_q = 0\). The conjecture states that if \(m_i\) is the number of terms in \(f_i\), then \((m_1-1)\cdots(m_k-1)\) is an upper bound. In the counter-example cases are considered with \(k=2\) and \(m_1 = m_2 = 3\) and it is proved that there exist \(5\) positive solutions. The author discusses possible alternative upper bounds.