A note on the sign degree of formulas

Abstract: Recent breakthroughs in quantum query complexity have shown that any formula of size n can be evaluated with O(sqrt(n)log(n)/log log(n)) many quantum queries in the bounded-error setting [FGG08, ACRSZ07, RS08b, Rei09]. In particular, this gives an upper bound on the approximate polynomial degree of formulas of the same magnitude, as approximate polynomial degree is a lower bound on quantum query complexity [BBCMW01]. These results essentially answer in the affirmative a conjecture of O'Donnell and Servedio [O'DS03] that the sign degree--the minimal degree of a polynomial that agrees in sign with a function on the Boolean cube--of every formula of size n is O(sqrt(n)). In this note, we show that sign degree is super-multiplicative under function composition. Combining this result with the above mentioned upper bounds on the quantum query complexity of formulas allows the removal of logarithmic factors to show that the sign degree of every size n formula is at most sqrt(n).