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Suppose,

$ P(x) := \sum_{n\geqslant r} {p_{n}x^n}$ and $ Q(x) := \sum_{n\geqslant s} {q_{n}x^n} $

are formal power series, where $p_{r}$ and $q_{s}$ doesnt equal $0$ such that $x^r$ is the smallest-order non-zero term of P(x) and $x^s$ is the lowest for Q(x).

By a Corollary stating: Let R(x) and S(x) be a formal power series. If the constant term of S(x) is non-zero, then there is a formal power-series C(x) such that $$ S(x)C(x) = R(x) $$ thus the solution A(x) is unique

From the corollary, $s=0$ is an okay condition such that Q(x)C(x) = P(x) to have a solution. I need to give a necessary condition for Q(x)C(x) = P(x) to have a solution, and prove that it is right. Also when a solution exists, is it unique?

Any hints to show how i can have a condition for this equation to have a solution?

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1 Answer 1

Show that $r\ge s$ is sufficient by considering the problem $Q_1(x) C_1(x) = P_1(x)$ with $Q(x)=x^sQ_1(x)$, $P(x)=x^r P_1(x)$. Show that it is also necessary because $x^s|Q(x)C(x)$.

For uniqueness, if $C(x)\ne D(x)$, let $x^t$ be the highest power dividing $C(x)-D(x)$ and observe that $x^{s+t}$ is the highest power dividing $Q(x)(C(x)-D(x))$, which thus cannot be the zero series, hence $Q(x)C(x)\ne Q(x)D(x)$-

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