Power roots of linearized polynomials

Probabilistic algorithm for finding roots of linearized polynomials

A probabilistic algorithm is presented for finding a basis of the root space of a linearized polynomial

Counting Roots of Polynomials Over Prime Power Rings

Suppose $p$ is a prime, $t$ is a positive integer, and $f\!\in\!\mathbb{Z}[x]$ is a univariate polynomial of degree $d$ with coefficients of absolute value $<\!p^t$. We show that for any fixed $t$, we can compute the number of roots in $\mathbb{Z}/(p^t)$ of $f$ in deterministic time $(d+\log p)^{O(1)}$. This fixed parameter tractability appears to be new for $t\!\geq\!3$. A consequence for arit...

Gröbner Bases for Linearized Polynomials

In this work we develop the theory of Gröbner bases for modules over the ring of univariate linearized polynomials with coefficients from a finite field.

Roots of Polynomials

A direct corollary of the fundamental theorem of algebra is that p(x) can be factorized over the complex domain into a product an(x − r1)(x − r2) · · · (x − rn), where an is the leading coefficient and r1, r2, . . . , rn are all of its n complex roots. We will look at how to find roots, or zeros, of polynomials in one variable. The solution of multi-variate polynomials can often be transformed ...

Polynomials , roots , and interlacing