A Family of Algorithms for Powering Sparse Polynomials

We discuss four new algorithms from a family of algorithms for computing integer powers of sparse polynomials. The four algorithms form a sequence of successively better algorithms; even the first member of the sequence shows an improvement in the leading term of the cost function in comparison with the best previously known binomial-expansion algorithm. To quote one result, if $f$ is a 32-term sparse polynomial, computing $f * f^9 $ takes $8.75 \times 10^9 $ multiplications, while the best new algorithm computes $f^{10} $ from scratch using $1.125 \times 10^9 $ multiplications. The time and space analyses given support the conjecture that the best new algorithm is optimal for time and space within the family of sequential binomial-expansion algorithms for this problem. If the input polynomial has $t$ terms, then the $n$th power may be computed by this algorithm with a time complexity of \[ \frac{{t^n }}{{n!}} + t^{n - 1} \left[ \frac{1}{{2(n - 2)!}} + \frac{1}{{2^{n - 2} (n - 1)!}}\right] + O(t^{n - 2} ),\quad n > 2 , \] and a space complexity of \[ \frac{{t^{n - 1} }}{{2^{2n - 4} (n - 1)!}} + O(t^{n - 2} ),\quad n > 2. \]