We are concerned with solving polynomial equations over rings. More precisely, given a commutative domain A with 1 and a polynomial equation antn + ...+ a0 = 0 with coefficients ai in A, our problem is to find its roots in A.
We show that when A = B[x] is a polynomial ring, our problem can be reduced to solving a finite sequence of polynomial equations over B. As an application of this reduction, we obtain a finite algorithm for solving a polynomial equation over A when A is F[x1, ..., xN] or F(x1, ..., xN) for any finite field F and any number N of variables.
The case of quadratic equations in characteristic two is studied in detail.
@article{urn:eudml:doc:41332,
title = {Solving quadratic equations over polynomial rings of characteristic two.},
journal = {Publicacions Matem\`atiques},
volume = {42},
year = {1998},
pages = {131-142},
mrnumber = {MR1628154},
zbl = {0915.13017},
language = {en},
url = {http://dml.mathdoc.fr/item/urn:eudml:doc:41332}
}
Cherly, Jorgen; Gallardo, Luis; Vaserstein, Leonid; Wheland, Ethel. Solving quadratic equations over polynomial rings of characteristic two.. Publicacions Matemàtiques, Tome 42 (1998) pp. 131-142. http://gdmltest.u-ga.fr/item/urn:eudml:doc:41332/