On the Computing Power of $+$, $-$, and $\times$

Abstract: Modify the Blum-Shub-Smale model of computation replacing the permitted computational primitives (the real field operations) with any finite set $B$ of real functions semialgebraic over the rationals. Consider the class of boolean decision problems that can be solved in polynomial time in the new model by machines with no machine constants. How does this class depend on $B$? We prove that it is always contained in the class obtained for $B = {+, -, \times}$. Moreover, if $B$ is a set of continuous semialgebraic functions containing $+$ and $-$, and such that arbitrarily small numbers can be computed using $B$, then we have the following dichotomy: either our class is $\mathsf P$ or it coincides with the class obtained for $B = {+, -, \times}$.