The Partial Derivative method in Arithmetic Circuit Complexity

In this thesis we survey the technique of analyzing the partial derivatives of a polynomial to prove lower bounds for restricted classes of arithmetic circuits. The technique is also useful in designing algorithms for learning arithmetic circuits and we study the application of the method of partial derivatives in this setting. We also look at polynomial identity testing and survey an e cient algorithm for identity testing certain classes of arithmetic circuits. The dimension of the vector space of partial derivatives of a polynomial will be the quantity of interest in this thesis. Lower bounds for an explicit polynomial against a class of circuits is obtained by comparing the dimension of the space of partial derivatives of the polynomial and the dimension of the space of polynomials computed by that class of circuits. The e ciency of the learning algorithms that we survey also depends on the dimension of partial derivatives of the class of functions. We also note the connections between the problems of proving lower bounds, designing e cient learning algorithms and identity testing arithmetic circuits.