How many ways can one cover $\mathbb Q_p$ with the images of polynomials? - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-20T04:19:37Z http://mathoverflow.net/feeds/question/112458 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/112458/how-many-ways-can-one-cover-mathbb-q-p-with-the-images-of-polynomials How many ways can one cover $\mathbb Q_p$ with the images of polynomials? Will Sawin 2012-11-15T06:22:01Z 2012-11-16T04:03:49Z <p>Define a finite set of polynomials over a field $K$ to <em>cover</em> $K$ if the images of the polynomials, viewed as functions from $K$ to itself, have union the whole set.</p> <p>Define a <em>minimal cover</em> to be a finite set of polynomials that cover a field, but such that no proper subset covers that field.</p> <blockquote> <p>Can you classify all minimal covers of $\mathbb Q_p$?</p> </blockquote> <p>A minimal cover of a number field must consist of just a single, linear polynomial. Indeed, for most $y$, for each polynomial $f$ in the set, $f(x)-y$ is an irreducible polynomial in $x$ by Hilbert irreducibility. $y$ is in the image of some $f$ so some irreducible polynomial $f(x)-y$ has a root, so is linear, so $f(x)$ is linear. A single linear polynomial covers the set so no other polynomials are needed. This is a special case of <a href="http://mathoverflow.net/questions/110544/is-there-any-theorem-like-implicit-function-theorem-in-mathbbq/110554#110554" rel="nofollow">this argument</a>.</p> <p>This does not hold in $\mathbb Q_p$. This is because <code>$\mathbb Q_p^\times/\left(\mathbb Q_p^\times\right)^n$</code> is finite, so if $S$ is a set of coset representatives for <code>$\mathbb Q_p^\times/\left(\mathbb Q_p^\times\right)^n$</code>, then <code>$\{ s x^n | s \in S\}$</code> is a finite set of nonlinear polynomials that cover $f$. It is also a minimal cover.</p> <blockquote> <p>Is this the only kind of minimal cover of $\mathbb Q_p$, up to translation and other obvious things?</p> </blockquote> <p>The only result I have in this direction is that if you take a cover of $\mathbb Q_p$, the leading terms of all the polynomials also cover $\mathbb Q_p$. This is because every coset of <code>$\left(\mathbb Q_p^\times\right)^n$</code> (where $n$ is a multiple of the degree of each polynomial) has elements which are very large in the $p$-adic norm, and when $f(x)$ is very large in the $p$-adic norm $x$ must be very large, and when $x$ is very large, the terms other than the leading term do not change what coset $f(x)$ is in, so there must be one polynomial whose leading term can reach each coset.</p> <p>A vaguely related question:</p> <blockquote> <p>What interesting fields, other than finite extensions of $\mathbb Q_p$, admit nontrivial/interesting minimal covers?</p> </blockquote>