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No, for rather trivial reasons. Consider the polynomial $f(X) = \varepsilon X - 1$ with $\varepsilon^2 = 0$, $\varepsilon \neq 0$, in $R = M_2(\mathbb{C})$. Then a root of $f(X)$ would mean that $\var …

Consider the complex projective variety given by $X^n = 0$, where $X\in \mathrm{M}_n(\mathbb{C})$ and, say, $n\geq 3$. Some basic properties of it are already mentioned in this question: https://mat …

Yes, there appears to be a follow-up work by Yansong Xu on the (99,66)-case: Intersection Numbers and the Jacobian Conjecture, in turn followed by The Jacobian Conjecture: Approximate roots and inters …

To (Q1): Yes, $F$ being invertible means in this context exactly that $F$ admits a polynomial compositional inverse. If you are working over a general commutative ring $k$, then invertibility of $J_F$ …

If $M$ denotes a $2n$-dimensional real smooth manifold, then $M$ together with $J\in\operatorname{End}(TM)$ with $J^2 =\mathrm{id}$ (also called almost product structure on $M$) have been surveyed i …

If $F:V\to W$ is a smooth at $a\in V$ function between finite-dimensional vector spaces over $\mathbb{R}$, then we have $$ F(x) = \sum_{k=0}^N\frac{1}{k!}(D^kF)(a)[(x-a)^{\otimes k}]+\text{remainder}, …

In [BP08] Poonen constructs and studies $\mathfrak{B}_n$, the moduli space of all based $n$-dimensional commutative associative unital $\kappa$-algebras, where $\kappa$ is an algebraically closed fiel …

To (Q1): Yes, see Differential Galois Theory. For example, the Galois group of the Airy equation over $\mathbb{C}$ is given by the Lie group $\operatorname{SL}_2(\mathbb{C})$. To (Q2): Yes, see (Q4). …

I am looking for further proofs, preferably in the literature, of the following result: Proposition: Let $R$ be a unital commutative $\mathbb{Z}$-torsion free ring. If $f(x) \in R[x]$ with $f'(x) \in …

I would suggest Altman and Kleiman's "A Term of Commutative Algebra". As far as I know, the most recent version can be found at https://dspace.mit.edu/handle/1721.1/116075.2. Quoting from the preface: …

Let $A$ be a commutative ring with unity. Denote $X:=Spec(A)$. $X$ is connected if and only if $A$ cannot be written as a Cartesian product $A_1\times A_2$ for nontrivial $A_1$ and $A_2$. So, that g …

A local ring $(A,\mathfrak{m})$ is regular iff the minimal number of generators of $\mathfrak{m}$ equals the Krull dimension of $A$. But Artinian rings have Krull dimension $0$, i.e. $\mathfrak{m}=0$, …

All rings are assumed commutative and unital. Some context (feel free to skip right to the questions below). I am trying to understand to what extent the property "being prime" for an ideal $I$ is int …

Let $(A,\mathfrak{m},\kappa)$ denote a commutative local Artinian ring. Somewhat by accident, I've stumbled across the following interesting decomposition: $$ A(\!(t)\!)^\times = t^\mathbb{Z} \cdot (1 …

I don't know if there is a more up-to-date precise statement for $\mathbb{F}_n$ uniformly for all $n \in \mathbb{N}$, my knowledge is definitely not recent, but here is what I know. Denote your LG-mod …