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Commutative rings, modules, ideals, homological algebra, computational aspects, invariant theory, connections to algebraic geometry and combinatorics.

10 votes
Accepted

If a formal power series over the complex numbers satisfies a polynomial identity, does it ...

The equation $\Phi(w,z)=0$ can be solved using Puiseux series. If $\frac{\partial{\Phi}}{\partial{w}}\not\equiv 0$ then there exist finitely many formal series $f(z)=\sum_{n\geq0}a_nz^{n/p}$ such t …
Loïc Teyssier's user avatar
6 votes
2 answers
263 views

Germs at infinity of sequence of integers

Consider the $\mathbb Z$-module $\mathcal Z$ obtained as the set of sequences of integers $\mathbb Z ^ \mathbb N$ modulo the relation that two sequences are deemed equivalent when their difference is …
Loïc Teyssier's user avatar
4 votes
1 answer
178 views

Effective bound on "Jacobian rank" for (regular) planar algebraic curves

Let an irreducible, square-free complex polynomial $f\in \mathbb C[x,y]$ be given. It is well known that the curve $\mathcal C:=\{f=0\}$ is nonsingular if and only if $\mathbb C[x,y]=<f,\partial_xf,\p …
Loïc Teyssier's user avatar
3 votes
1 answer
279 views

Composite families of formal power series over $\mathbb C$ as algebraic variety

I was led to prove that the set of composite families $(f_j)_{j \leq k}$ of germs at $0\in \mathbb C^m$ of a holomorphic function (composite = sharing a common divisor belonging to the maximal ideal) …
Loïc Teyssier's user avatar
2 votes
Accepted

Composite families of formal power series over $\mathbb C$ as algebraic variety

For those interested in the question, see my paper on the subject http://fr.arxiv.org/abs/1308.6371v2 , section 6.
Loïc Teyssier's user avatar
2 votes

What does a singular simplex with real coefficient mean

It's only a formal (i.e. symbolic) sum, understood as an element of the $\mathbb R$-module generated by the simplexes $\sigma_i$ (similar to vectors from a basis in linear algebra), where addition is …
Loïc Teyssier's user avatar
2 votes
0 answers
65 views

Decidability of the solvability of quadratic systems

Let a finite collection of (complex) unknowns $\{x_1,\ldots,x_n\}$ be given, as well as an affine system $AX=B$ in the quadratic variables $X:=[x_i x_j : i\leq j]$, with entries in a computable subfie …
Loïc Teyssier's user avatar