New bounds for the number of connected components of fewnomial hypersurfaces

Bihan Frédéric, Humbert Tristan, Tavenas Sébastien

Published 2022-08-09Version 1

We prove that the zero set of a \(5\)-nomial in \(n\) variables, whose exponent vectors are not colinear, has at most $\lfloor \frac{n-1}{2}\rfloor +3$ connected components in the positive orthant. Moreover, we give an explicit \(5\)-nomial in $2$ variables which defines a curve with three connected components in the posititive orthant, showing that our bound is sharp for $n=2$. In a more general setting, if \(\A = \{a_0,\ldots,a_{d+k}\} \subset \Z^n\) has dimension \(d\), we prove that the number of connected components of the zero set of a \(\A\)-polynomial in the positive orthant is smaller than or equal to \(18 (d+1)^{k-1} 2^{\binom{k-1}{2}} + k+1\), improving the previously known bounds. Moreover, our results continue to work for polynomials with real exponents.