Any proof of P!=NP will have to overcome two barriers: relativization
and natural proofs. Yet over the last decade, we have seen circuit
lower bounds (for example, that PP does not have linear-size circuits)
that overcome both barriers simultaneously. So the question arises of
whether there is a third barrier to progress on the central questions in
complexity theory.

In this paper we present such a barrier, which we call algebraic
relativization or algebrization. The idea is that, when we relativize
some complexity class inclusion, we should give the simulating machine
access not only to an oracle A, but also to the low-degree extension of
A over a finite field or ring.

We systematically go through basic results and open problems in
complexity theory to delineate the power of the new algebrization
barrier. First, we show that all known non-relativizing results based
on arithmetization -- both inclusions such as IP=PSPACE and MIP=NEXP,
and separations such as MA-EXP not in P/poly -- do indeed algebrize.
Second, we show that almost all of the major open problems -- including
P versus NP, P versus RP, and NEXP versus P/poly -- will require
non-algebrizing techniques. In some cases algebrization seems to
explain exactly why progress stopped where it did: for example, why we
have superlinear circuit lower bounds for PromiseMA but not for NP.

Our second set of results follows from lower bounds in a new model of
algebraic query complexity, which we introduce in this paper and which
is interesting in its own right. Some of our lower bounds use direct
combinatorial and algebraic arguments, while others stem from a
surprising connection between our model and communication complexity.
Using this connection, we are also able to give an MA-protocol for the
Inner Product function with O(sqrt(n) log(n)) communication (essentially
matching a lower bound of Klauck), as well as a communication complexity
conjecture whose truth would imply NL!=NP.