If I understand this correctly, arithmetizating a formula is a way of "interpolating" a polynomial in such a way that polynomial evaluated on 0's and 1's corresponds to "unsatisfiable" if it is a root, and "satisfiable" if it is 1. (I use interpolating in a very loose sense here). It seems plausible that the degree/number of turning points of the resulting polynomial corresponds loosely to how "complex" the formula is.
To give a trivial example,
$a \oplus b$ is quardratic, $(1-ab)(1-(1-a)(1-b))$
$a \wedge b$ is $ab$, linear.
I believe an argument of this sort was used by Smolensky to show that PARITY is not in $AC^0$. I wonder, however, if there is there a more precise interpretation of
- The Degree of the Polynomial
The Number of Turning Points in the Polynomial?