Finding a non-convergent Cauchy sequence

[Euler2718]

TL;DR Summary

Under a metric defined on the set of single variable real valued coefficient polynomials, find a non-convergent Cauchy sequence.

Define a metric on $\mathbb{R}[x]$ for distinct polynomials $f(x),g(x)$ as $d(f(x),g(x)) = \frac{1}{2^{n}}$, where $n$ is the largest positive integer such that $x^{n}$ divides $f(x)-g(x)$. Equivalently, $n$ is the multiplicity of the root $x=0$ of $f(x)-g(x)$. Set $d(f(x),f(x))=0$.

Goal: Show $(\mathbb{R}[x],d)$ is not complete via counter-example.

Divisbility has never been my strong suit. Most of my thinking has been based off of the assumption that the value of $n$ is proportional to the degree of $f(x)-g(x)$. I first considered a sequence $(\mathcal{F}_{k})_{k=0}^{\infty}$ where $\deg(\mathcal{F}_{0})=0$ and $\deg(\mathcal{F}_{k+1})=\deg(\mathcal{F}_{k})+1$ . Based on my assumption of how $n$ behaves though, this seems to converge as $n$ gets larger and larger. I then considered a sequence where the degrees are the same throughout, but this would not be Cauchy as the $n$ would be the same for all $\mathcal{F}_{k}$. Then I thought about sequences where the degree decreases, but since the degree is a non-negative integer, it would eventually just turn into a sequence of zero-degree polynomials, which wouldn't work.

So I'm starting to think I've made an error in my assumption about how $n$ works, or that maybe looking at the multiplicity route may be more fruitful (though I cannot see at the moment how). Any input would be appreciated.

 

[member 587159]

An idea (not sure it works)

Consider the sequence
$$\left\{\sum_{k=0}^n \frac{x^k}{k!}\right\}_{n=0}^\infty$$
Then for $p < q$ we have
$$d\left(\sum_{k=0}^q \frac{x^k}{k!}, \sum_{k=0}^p \frac{x^k}{k!} \right) = 1/2^{p+1}$$
and then it is evident that this is a Cauchy sequence.

My intuition here is that under the sup-metric on the polynomials, this sequence wants to converge to $e^x$ which is not a polynomial. So maybe we can hope that happens here as well.

So maybe you can show this does not converge?

EDIT: Maybe even simpler: you can try to show $\{1,1+x, 1+x+x^2, \dots\}$ is Cauchy but does not converge

 

[Infrared]

@Math_QED

It works. Let $p_n$ be the $n$-th polynomial in your sequence. Let $p$ be a polynomial of degree $N$. Since $p$ has no term of degree $N+1$ but $p_n$ does when $n\geq N+1$, we find that $x^{N+1}$ is the highest power of $x$ that can divide $p-p_n$, so $|p-p_n|\geq 2^{-N-1}.$ This means the sequence $(p_n)$ cannot converge to any polynomial of any degree $N$.

There's no need for the factorials; $p_n=1+x+\ldots x^n$ works fine too.

 

[Euler2718]

Thank you both.

 

[mathwonk]

This example is the reason for the existence of this concept. I.e. one wants to mimic the construction of power series within algebra. Thus power series are just the completion of the polynomials in an appropriate metric. But abstracted this way, it also generalizes to many other rings, with topology defined by powers of an ideal. In abstract algebraic geometry, where the Zariski topology is defined by the sets where polynomials fail to vanish, these open sets are too large for fine structure analysis. To focus more closely on the neighborhood of a point, one can consider completions of the Zariski local rings. A Zariski local property is something reflected in the Zariski local ring of the point, whereas an analytically local property is something reflected by the complete local ring of the point. A point on a variety is a smooth, or manifold - like, point for example, if that completion is (isomorphic to) a power series. (Actually for this property of a point, "simplicity", the Zariski local ring also reflects it by the fact that the Zariski local ring at a simple point is "regular".)

https://en.wikipedia.org/wiki/Completion_of_a_ring

One interesting and perhaps surprizing thing about power series rings is they have more units than polynomial rings, i.e. many polynomials have power series inverses. E.g. the polynomial (1-x) has as (multiplicative) inverse the power series 1+x+x^2+x^3+... I think that is really cool, and it is also useful. (One fun consequence of this is it makes Euclid's proof that there are infinitely prime integers, fail for power series, since x is prime but 1+x is not.)

An example of the extra insight granted by the completion of a local ring in geometry is the case of a figure eight, which has two branches crossing at the center point. In a small analytic nbhd of this point, the curve becomes reducible, i.e. breaks into two pieces, but every Zariski nbhd is still irreducible, i.e. has only one piece. Correspondingly, the Zariski local ring at this point is a domain, i.e. has no zero divisors, but the completion of that ring does have zero divisors, reflecting the local analytic reducibility. I.e. analytically locally, there is a local function vanishing on one branch and not the other, while the product of two such functions, one vanishing on each branch, is dead zero, at least on an analytic local nbhd. The point seems to be that a polynomial vanishing on one branch would vanish on the whole curve, but a power series, with smaller domain, can vanish on just one branch.

 

[zinq]

Can't it even be a simpler sequence, like {P_n} defined by P_n(x) = xⁿ ?

Then for any ε > 0, just pick N = 1 + floor(log₂(ε)) and it follows that for m, n ≥ N we have
d(x^m - x^n) = 1/2^min(m,n) < 1/2^N < ε.

So this is a Cauchy sequence as well.

 

[Infrared]

Yes, as long as the $x^k$ term is eventually constant for each $k$, the sequence is Cauchy (and vice-versa).

 

[member 587159]

Can't it even be a simpler sequence, like {P_n} defined by P_n(x) = xⁿ ?

Then for any ε > 0, just pick N = 1 + floor(log₂(ε)) and it follows that for m, n ≥ N we have
d(x^m - x^n) = 1/2^min(m,n) < 1/2^N < ε.

So this is a Cauchy sequence as well.

This sequences converges to 0 though.

 

[zinq]

Good point, since 0 is a polynomial. So this trivial example of powers of x does not show the space of polynomials to be incomplete.

 

[mathwonk]

Right. Notice that it is the differences between elements of a sequence, that approach zero in a Cauchy sequence. This sequence of monomials x^k, is in a sense a sequence of differences for the sequence given in post #3 above. what do you think is the "completion" of that space, with the given metric? Can you get every power series? I.e. do all power series "converge" in this metric? Can you get anything else?

 

[zinq]

It appears to me that with the metric defined in #1, the Cauchy completion of the ring of real polynomials is the ring of formal power series over the reals. Not immediately sure if this works for an arbitrary field replacing the reals here.