Support of a formal series

Definition

Let \(\mathbb{K}\) be a field and \(t\) a variable. Let us consider \(y(t) = \sum_{p \in \mathbb{N}} c_p t^p \in \mathbb{K}((t))\). It is defined the support of \(y(t)\) as

\[ Supp(y(t)) = \{p ~ | ~ c_p \ne 0 \}. \]

Examples

\(\circ\) Let \(\mathbb{K}\) be a field, \(t\) and indeterminant and the rings \(\mathbb{K}[t], \mathbb{K}[[t]]\). By definition of polynomial ring, it holds that

\[ \mathbb{K}[t] = \{y(t) \in \mathbb{K}[[t]] ~ | ~ Supp(y(t)) < + \infty \} \subsetneq \mathbb{K}[[t]]. \]

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.