Decision trees: old and new results

In this paper, we prove two general lower bounds for algebraic decision trees which test membership in a set $S\subseteq R^n$ which is defined by linear inequalities. Let $rank(S)$ be the maximal dimension of a linear subspace contained in the closure of $S$ {in the Euclidean topology}. First we show that any decision tree for $S$ which uses products of linear functions (we call such functions {\em mlf-functions}) must have depth at least $n-rank(S)$. This solves an open question raised by A.C.~Yao and can be used to show that mlf-functions are not really more powerful than simple comparisons between the input variables when computing the largest $k$ out of $n$ elements. Yao proved this result in the special case when products of at most two linear functions are allowed. Out proof also shows that any decision tree for this problem must have exponential size. Using the same methods, we can give an alternative proof of Rabin's Theorem, namely that the depth of any decision tree for $S$ using arbitrary analytic functions is at least $n-rank(S)$