# Non-linear convolution and power series

1. Aug 27, 2013

### burritoloco

1. The problem statement, all variables and given/known data

Hi, suppose we have the summation
$$\sum_{i=0}^{n-1} \sum_{j=0}^{n-1} a_j b_{i-j}^{j} x^i,$$

where the subscripts are taken modulo $n$, and $a_i^n = a_i, b_i^n = b_i$ for each $i$.

Write the above power series as a product of two power series modulo $x^n - x$.

2. Relevant equations

I'm only aware of the regular Cauchy (linear) convolution. That is,
$$\sum_{i=0}^{n-1}\sum_{j=0}^{n-1}a_j b_{i-j}x^i = \left( \sum_{i=0}^{n-1}a_i x^i \right) \left( \sum_{j=0}^{n-1}b_j x^j \right).$$

3. The attempt at a solution
I'm frankly not sure...

Thanks!

Last edited: Aug 27, 2013