# Given a formal power series ,decide whether there exists a polynomial the series satisfies and if it exists,how to write it down?

Given a formal power series $$y(x)=\sum_{i=0}^{\infty} a_i x^i$$ Is there an algorithm that decides whether there exists a polynomial$$P(x,y)=p_n(x)y^n+p_{n-1}(x)y^{n-1}+\cdots+p_0(x)=0,p_j(x)\in F[x]$$the series satisfies and if it exists,how to write it down?This is relevant because if an element of $\mathbf{Z}[[X]]$ is algebraic over
$\mathbf{Q}(X)$, then its reduction modulo any prime $p$ will be algebraic
over $\mathbf{F}_p(X)$.

# Loday's characterization and enumeration of faces of associahedra (Stasheff polytopes)

From "The multiple facets of the associahedra" by Loday:

Let us consider the formal power series

$$f(x) = x+a_1 x^2 +a_2 x^3 + \cdots+ a_n x^{n+1} + \cdots$$

and let

$$g(x) = x+b_1 x^2 + b_2 x^3 + \cdots + b_n x^{n+1} + \cdots$$

be its inverse for composition, that is,

$$f(g(x)) = x.$$

The coefficient $b_n$ is a polynomial in $a_1$ to $a_n$.

# Prove a family of series having integer coefficients

I encountered a certain family of infinite series in some work, which is given by
$$F_r(x)=\frac1{2^r}\sum_{k=0}^r\binom{r}k\frac1{1+x(2k-r)^2}.For each r\in\mathbb{N}, the Taylor series for F_r(x) has integer coefficients. # Principal ideal subrings of formal power series rings Observe that g(x_1+x_2, f(x_1) + f(x_2))= g(x_1+x_2, f(x_1+x_2)) vanishes as an element of \mathbb F[[x]] \otimes \mathbb F[[x]].So it vanishes as an element of R \otimes R = F[x,f]/g(x,f) \otimes F[x,f]/g(x,f). # Notation and reference for polynomials with coefficients not commuting with the indeterminates Then one constructs the ring of "fully noncommutative" polynomials in X in the obvious way.Analogously, one gets "fully noncommutative" polynomials and formal power series over R in several or infinitely many noncommutative variables. # Coherence of subrings of K[[X,Y]] Let K[[X,Y]] be a two-variables formal power series ring over a field K.Consider a sub-ring \iota \colon A \subset K[[X,Y]]. # Flat Quotients of Power Series Rings I read the following statement in some algebraic topology notes and I want to know if it is true and, if so, why.Let R be a commutative ring and f(x) a power series in R[[x]]. # Generating function of the Thue-Morse sequence Let T be the generating function of the Thue-Morse sequence; thus, T(x)=x+x^2+x^4+x^7+\dotsb.It is known that T satisfies the nice congruence$$ (1+x)^3 T^2(x) + (1+x)^2 T(x) + x \equiv 0 \pmod 2 $$(the congruence is actually modulo the principal ideal generated by 2 in the ring of formal power series {\mathbb Z}[[x]]). # Are hyperreal numbers isomorphic to formal power series? For instance, e^{\omega}=\frac{\omega^0}{0!f(x)=x corresponds to \omega f(x)=x^2 corresponds to \omega^2 etc. # For which recurrence relations is it decidable whether a formal power series has a maximal zero coefficient? In this MSE question, I asked whether one can prove that a generating function has infinitely many coefficients equal to zero.For which R is it decidable whether G has a maximal zero coefficient a_m=0? # Smoothness in Ecalle's method for fractional iterates Does the formal power series solution to f(f(x))= \sin( x)  converge?Yesterday I did the same thing for x + x^2, where the conclusion is analyticity for x > 0 and continuity at 0. # existence and uniqueness of solutions for ODEs in formal power series? There is a formal differentiation in the ring of formal power series k[[x]].Let F(x,y) \in k[[x,y]] be a formal series which is algebraic over x and y. # Exponential of a specific hypergeometric series Let U(x,y,z) be a formal power series with constant term 1 and integer coefficients.Let U(x,y,z) be a formal power series with constant term 1 and integer coefficients. # Can we prove that the ring of formal power series over a noetherian ring is noetherian without axiom of choice? Can we prove that every non-empty set of ideals of A[[x]] has a maximal element without Axiom of Choice?The set of ideals \{I_{\alpha}^{(\omega)}\}_{\alpha} has a maximal element, say I^{(\omega)}_{\beta}. # Identifying a special function from its power series Here n,p,r are integers with n\ge 0 and p\ge r\ge 0:$$
f_{n,p,r}(x)\,=\,\sum_{s=0}^p \frac{x^s}{s!The answer is likely zero if $r<p$ and reasonably simple if $r=p$, but how to prove that?

# Are these powers of a characteristic 3 power series annihilated by certain Hecke operators?

(So D=x+2(x^7)+2(x^13)+2(x^19)+x^25+(higher degree terms).)

There are formal Hecke operators T_p: Z/3[[x]]-->Z/3[[x]] for all primes p other than 3.