Cancellation is Exponentially Powerful for Computing the Determinant

By de ning a notion of cancellation in algebraic circuits, we study the power of negation at a ner granularity than previously considered. Our result is that in the absence of cancellation, computing the determinant requires exponential circuit size. Previous work shows that cancellations are essential for e ciently computing certain monotone (all coe cients positive) polynomials. The work presented in this paper extends that to the computation of non-monotone polynomials and shows that it is the cancellative aspect of negation that allows e cient computation of even certain non-monotone polynomials. We present the rst example of a non-monotone polynomial for which cancellations lead to exponential savings in circuit size: determinant.