Root continuity principle in $\overline{\mathbb{C}((t))}[T]$.

Question

is anyone aware of an extension of the usual argument for root continuity for polynomials with complex coefficients to the case where the base field is the Puiseux series field over the complex ? Here, I mean continuity for the product of complex topology (i.e.) coefficients to cefficients. In fact any extension containing the ring of formal power series would be fine for my purpose.

Thanks in advance.

Answer 1

• You didn't specify your question clearly, but it is easy to show that the roots move continuously with the coefficients, continbuously for the usual non-archimedian valuation on formal series, when looking at the monic polynomials of a fixed degree. With $A=\bigcup_{n\ge 1}\Bbb{C}[[x^{1/n}]]$, the valuation $v$ on $A$ extends to a valuation $w(\sum_{j=0}^d a_j T^j) = \inf_j v(a_j)$ on $A[T]$, factorizing $f(T)=\prod_{j=1}^d (T-\alpha_j)$, then $v(z) \le d \sup_j v(z-\alpha_j)$, and if $w(g(T)-f(T))> \sup_j v(z-\alpha_j)$ then $v(g(z)) < v(g(z)-f(z))$ so that $v(g(z))=v(f(z))$ ie. $g(z)\ne 0$.

• The roots also move continuously with the coefficients, this time for the product of complex topology, when looking at the polynomials $\in \Bbb{C}[[x]][T]_{monic}$ of a fixed degree whose reduction $\bmod x$ is separable.

  Separable $\bmod x$ (with Hensel lemma) ensures that the polynomial splits completely in $\Bbb{C}[[x]]$. Product of complex topology means the topology given by the semi-norms $\|\sum_{n\ge 0} c_n x^{n/m}\|_k = \sup_{n/m\le k} |c_n|$. It extends naturally to some semi-norms on the polynomials of $\Bbb{C}[[x]][T]$ of a given degree.

  It is mostly immediate (for separable $\bmod x$ polynomials) from the Hensel lemma construction of the roots that this is continuous in the coefficients in the complex topology.

  Why do we need the separable condition? Because the roots of $T^m-x$ are the $e^{2i\pi l/m} x^{1/m}$, and there are some polynomials $\in \Bbb{C}[[x]][T]$ arbitrary close to $T^m-x$ in product of complex topology whose reduction is separable $\bmod x$, ie. that split completely in $\Bbb{C}[[x]]$, so their roots can't be close to the $e^{2i\pi l/m} x^{1/m}$ in complex topology.