### Abstract:

Skew polynomial rings F[x; σ, δ] with coefficients over a division ring F (Definition 1.1.6),
were introduced in [27] by Oystein Ore (1933), as a non-commutative generalization of
the conventional polynomial rings. The first applications of skew polynomials appear
with the work of [9, 10] and Jacobson [18] and recently, they have been used to construct
algebraic codes (e.g. see [4, 6, 25]) and for applications in cryptography [5].
Although in general F[x; σ, δ] behaves differently from the classical polynomial ring, it
preserves the important property of having a Euclidean division algorithm. However, this
algorithm holds for right division and not for left division, unless σ is an automorphism
of F, as stated in [27, Theorem 6]. This property, allowed Lam and Leroy in [21, p. 310]
to define the evaluation of a polynomial f(x) ∈ F[x; σ, δ] at any point a ∈ F, as the
unique remainder of the right-hand division of f(x) by x − a (Definition 1.1.11), forcing
a remainder theorem as in the classical case. Having an evaluation map is key to begin
the study of the zeros of a skew polynomial, but unlike the classical case, this study is
more difficult, since in general a skew polynomial of degree n ≥ 2 can have more than n
zeros, possibly infinite (Example 1.1.15).
On the other hand, in literature there exist multivariate generalizations of F[x; σ, δ],
for instance the iterated polynomial rings F[x1, σ1, δ1][x2, σ2, δ2] · · · [xn, σn, δn] (see [30],
[9, p. 532]). However, to define an evaluation map that allows one to evaluate any
polynomial F ∈ F[x1, σ1, δ1][x2, σ2, δ2] · · · [xn, σn, δn] is not possible, because in general
unique remainder algorithms do not hold for iterated skew polynomials (see [23] for
more details). In 2019, the authors in [23] overcome this obstacle by considering an
alternative construction and introduce the so-called free multivariate skew polynomial
rings F[x1, x2, ..., xn; σ, δ] (Definition 1.2.3), showing that in this case, it is possible to
define the evaluation of any free skew polynomial F at any point (a1, a2, ..., an) ∈ F
n , as the unique remainder of the Euclidean division on the right of F by the polynomials
x1 − a1, x2 − a2, ..., xn − an (Definition 1.2.5).