  We prove lower bounds on the length of regular expressions for finite
languages by methods from arithmetic circuit complexity. First, we show a
reduction: the length of a regular expression for a language $L\subseteq
\{0,1\}^n$ is bounded from below by the minimum size of a monotone arithmetic
formula computing a polynomial that has $L$ as its set of exponent vectors,
viewing words as vectors. This result yields lower bounds for the binomial
language of all words with exactly $k$ ones and $n-k$ zeros and for the
language of all Dyck words of length $2n$. We also determine the blow-up of
language operations (intersection and shuffle) of regular expressions for
finite languages. Second, we adapt a lower bound method for multilinear
arithmetic formulas by so-called log-product polynomials to regular
expressions. With this method we show almost tight lower bounds for the
language of all binary numbers with $n$ bits that are divisible by a given odd
integer $p$, for the language of all words of length $n$ over a $k$ letter
alphabet with an even number of occurrences of each letter and for the language
of all permutations of $\{1,\dots, n\}$.
