A Lower Bound for the Formula Size of Rational Functions
Kyriakos Kalorkoti · SIAM Journal on Computing · 1985
We establish a lower bound for the formula size of quolynomials over arbitrary fields. Our basic formula operations are addition, subtraction, multiplication and division. The proof is based on Nečiporuk’s [Soviet Math. Doklady, 7 (1966), pp. 999–1000] lower bound for Boolean functions and uses formal power series. This result immediately yields a lower bound for the formula size of rational functions over infinite fields. We also show how to adapt Nečiporuk’s method to rational functions over finite fields. These results are then used to show that, over any field, the $n \times n$ determinant function has formula size at least $\Omega (n^3 )$. We thus have an algebraic analogue to the $\Omega (n^3 )$ lower bound for the Boolean determinant due to Kloss [Soviet Math. Doklady, 7 (1966), pp. 1537–1540].