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I'm curious as to whether there is a ring homomorphism from $\mathbb{Z} [[X]]$,the ring of formal power series in indeterminate $X$, to the rationals such that for every polynomial $p(X)$ in the ring of power series, $\phi(p(X))=p(3/4)$.

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  • $\begingroup$ Yes I mean that p is a formal polynomial $\endgroup$ Feb 21, 2020 at 2:13
  • $\begingroup$ Right. Sorry I persisted in being ambiguous hahah $\endgroup$ Feb 21, 2020 at 2:39

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The central binomial coefficients have generating function $p(X)=\sum_{n=0}^{\infty} \binom{2n}n X^n$ which satisfies $p(X)^2(1-4X)=1$ in $\mathbb Z[[X]].$

So a $\phi$ is a homomorphism sending $X\mapsto \frac{3}{4}$ would have to have $\alpha=\phi(p(X))$ satisfying $$\alpha^2 \left(1-4\frac{3}{4}\right)=1$$ Or $\alpha^2=\frac{-1}{2}.$ There is no such $\alpha,$ rational or real.

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The generating function $F \in \mathbb{Z}[[X]]$ of Catalan numbers $(C_n)_{n=0}^\infty$ is given by $$F(X) = \sum_{n=0}^\infty C_nX^n = \frac{1-\sqrt{1-4X}}{2X}$$ and hence it satisfies $$(1-2XF(X))^2 = 1-4X.$$ If such a homomorphism $\phi$ would exist, we would have $$(1-2\phi(X)\phi(F(X)))^2 = 1-4\phi(X) \implies \left(\underbrace{1-\frac32\phi(F(X))}_{\in\mathbb{Q}}\right)^2 = 1-4\cdot \frac34 = -2$$ which is a contradiction.

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