2
$\begingroup$

Consider a multivariable power series with positive coefficients such that it is known to converge on a $C^\infty$ (bounded) curve of $\mathbb{R}^n$, where $n$ is the number of variables. In addition, we know that the values this series takes on this curve are strictly less than 1, and that they are strictly positive on at least a segment of the curve and null on its extremities (which should imply that curve is at least inside the interior of the domain of convergence for some part of it if I'm not mistaken).

Is there anything we can say about the continuity and smoothness of the power series on this curve?

It seems to me that, for $n = 1$, we can prove the continuity on the radius of convergence in these same conditions, and that the same rationale can be used in the general case if we assume that the curve is not out of the interior of the domain of convergence of the power series except on a finite number of points (but I haven't properly proved it yet). But can we hope for some nice properties without any additional assumption on the curve and just what we know of the power series?

Edit: As I added thanks to Alexandre Eremenko's suggestion, I realized that the extremal points of the curves are of the form x1 = (1,...,1,0,...,0) and x2 = (0,...,0,1,...,1) and the power series is equal to 0 on a segment from x1 to another point of the curve.

Edit 2: I also forgot to point out that all the points of the curve other than x1 and x2 have their coordinates in [0,1)!

Edit 3: Positivity of the power series on a part of the curve included in the description (which would imply that the curve is at least partly in the interior of the domain of convergence considering that this part is between two zeros of the power series).

Improve this question
$\endgroup$
6
  • 1
    $\begingroup$ For the answer it is necessary to know how your curve is situated with respect to the origin. $\endgroup$
    – Alexandre Eremenko
    Commented Dec 18, 2016 at 20:42
  • $\begingroup$ It seems that I actually add some conditions, even though I don't know if it will give you this information: $\endgroup$
    – Raphael B
    Commented Dec 18, 2016 at 20:49
  • $\begingroup$ The extremal points of the curves are of the form x1 = (1,...,1,0,...,0) and x2 = (0,...,0,1,...,1) and the power series is equal to 0 on a segment from x1 to another point of the curve. $\endgroup$
    – Raphael B
    Commented Dec 18, 2016 at 20:52
  • $\begingroup$ I also forgot to point out that all the points of the curve other than x1 and x2 have their coordinates in [0,1)! $\endgroup$
    – Raphael B
    Commented Dec 18, 2016 at 21:48
  • $\begingroup$ But, putting n=2 for simplicity, in general convergence in (0,1) and (1,0) do not imply e.g. convergence in $(\epsilon,\epsilon)$. Consider a series $\sum_{k=0}^\infty c_k(xy)^k$. (so it will not guarantee that points of the curve are in the interior of the domain of convergence, besides the case of (t,0) and (0, t) for |t|<1) $\endgroup$
    – Pietro Majer
    Commented Dec 18, 2016 at 22:07

0

Reset to default

You must log in to answer this question.