On the Alekhnovich–Razborov degree lower bound

Polynomial calculus is a Hilbert-style proof system in which lines are polynomials modulo x = x (for each variable x) and the rules allow deriving c1P1 + c2P2 from P1, P2 and xP from P for a variable x. A polynomial calculus refutation of a set of axioms is a derivation of 1 from these axioms. Research in proof complexity tends to concentrate on the length of proofs. We will rather be interested in the degree of a refutation, which is the largest degree of a polynomial in the refutation. The degree of a polynomial calculus refutation is analogous to the width of a resolution proof, a quantity which has proved useful in obtaining length lower bounds [BSW01]. Although no such connection is known for polynomial calculus, it is still interesting to prove lower bounds on the degree required to refute certain sets of axioms. One of the few general techniques for proving degree lower bounds is due to Alekhnovich and Razborov [AR03]. The purpose of this note is to present an alternative proof of their lower bound. We illustrate the technique with two applications, refutations of random CNFs and refutations of the graph ordering principle over a random graph. The former appears already in [AR03], and the latter is due to Galesi and Lauria [GL10]. Section 3 provides background from commutative algebra. We formally introduce polynomial calculus in Section 2. We prove the main lower bound in an abstract framework in Section 4. In Section 5, we instantiate the framework for CNFs whose defining graph is a unique expander, following Alekhnovich and Razborov [AR03]. In Section 6, we instantiate the framework for the graph ordering principle over a unique expander, following Galesi and Lauria [GL10].