## Lower Bounds For The Polynomial Calculus (1998)

Citations: | 49 - 5 self |

### BibTeX

@MISC{Razborov98lowerbounds,

author = {Alexander A. Razborov},

title = {Lower Bounds For The Polynomial Calculus},

year = {1998}

}

### Years of Citing Articles

### OpenURL

### Abstract

We show that polynomial calculus proofs (sometimes also called Groebner proofs) of the pigeonhole principle PHP n must have degree at least (n=2)+1 over any field. This is the first non-trivial lower bound on the degree of polynomial calculus proofs obtained without using unproved complexity assumptions. We also show that for some modifications of PHP n , expressible by polynomials of at most logarithmic degree, our bound can be improved to linear in the number of variables. Finally, we show that for any Boolean function f n in n variables, every polynomial calculus proof of the statement "f n cannot be computed by any circuit of size t," must have degree t=n). Loosely speaking, this means that low degree polynomial calculus proofs do not prove NP 6 P=poly.