Expansion with respect to a set of polynomials

Definition

Let \(g,G_1, \ldots, G_h \in \mathbb{K}((x))[y]\) with \(G_1, \ldots, G_h\) non-zero and let \(f(x,y) = y^n + a_1(x)y^{n-1} + \cdots + a_n(x) \in \mathbb{K}((x))[y]\) be an irreducible polynomial. The definition of expansion of \(g\) with respect to \((G_1, G_2, \ldots, G_h, f)\) comes from the following result.

Proposition

Let \(g,G_1, \ldots, G_h \in \mathbb{K}((x))[y]\), with \(G_1, \ldots, G_h\) non-zero and let \(f(x,y) = y^n + a_1(x)y^{n-1} + \cdots + a_n(x) \in \mathbb{K}((x))[y]\) be an irreducible polynomial. Then,

\[ g = \sum_{\theta \in \Theta} c_{\theta} (x) G_1^{\theta_1} G_2^{\theta_2} \cdots G_h^{\theta_h} f^{\theta_{h+1}}, \]

for some \(\theta = (\theta_1, \ldots, \theta_{h+1}) \in \mathbb{N}^{h+1}\), with \(0 \le \theta_k < e_k\) for all \(k \in \{1, 2, \ldots, h\}\), where \((e_1, e_2, \ldots, e_h)\) comes from the definition of characteristic sequences of a formal series, and some \(c_{\theta} (x) \in \mathbb{K}((x))\). Moreover,

\[ deg_y g = \max \{deg_y ~~ c_{\theta} (x) G_1^{\theta_1} G_2^{\theta_2} \cdots G_h^{\theta_h} f^{\theta_{h+1}} ~ | ~ \theta \in \Theta \}. \]

It is defined the expansion of \(g\) with respect to \((G_1, G_2, \ldots, G_h, f)\) as the expression of \(g\) obtained in the previous proposition.

Examples

\(\circ\) Let \(g = y^{10}\), and \(f = y^6 - 2x^2y^3 + x^4 - x^5y, G_1 = y, G_2 = y^3 - x^2\), let us compute the expansion of \(g\) with respect to \((G_1, G_2, f)\). If we divide \(g\) by \(f\),

\[ g = (2x^2y + y^4)f + (2x^7y^2 - 2x^6y + x^5y^5 + 3x^4y^4) = Qf + R. \]

Since \(deg_y Q < deg_y f\), we start dividing \(Q\) and \(R\) by \(G_2\).

\[ Q = yG_2 + 3x^2y, \hspace{0.3cm} R = (x^5y^2 + 3x^4y)G_2 + 3x^7y^2 + x^6y. \]

Finally,

\[ g = (yG_2 + 3x^2y)f + (x^5y^2 + 3x^4y)G_2 + 3x^7y^2 + x^6y \]

\[ = G_1G_2f + 3x^2G_1f + x^5G_1^2G_2 + 3x^4G_1G_2 + 3x^7G_1^2 + x^6G_1. \]

References

Assi, Abdallah, Marco D’Anna, and Pedro A. Garcı́a-Sánchez. 2020. Numerical Semigroups and Applications. Vol. 3. RSME Springer Series. Springer, [Cham]. https://doi.org/10.1007/978-3-030-54943-5.