A vectorial approach to generalize the remainder theorem

Abstract

We propose a new computational proof for the division algorithm that, using vector algebra, generalizes the remainder theorem to divisions for polynomials of any degree over a generic integral domain. Then, we extend this result to calculate the pseudo-divisions.

Later, starting from the previous theorems, we obtain some algorithms that calculate the pseudo-remainder and the pseudo-quotient while avoiding long division.

Finally, we provide examples and comparisons indicating that these algorithms are efficient in divisions by sparse polynomials and their divisors, as cyclotomic polynomials.