On the degree of univariate polynomials over the integers

We study the following problem raised by von zur Gathen and Roche [GR97]: <i>What is the minimal degree of a nonconstant polynomial f</i>: {0,...,<i>n</i>} → {0,...,<i>m</i>}<i>?</i> Clearly, when <i>m</i> = <i>n</i> the function <i>f</i>(<i>x</i>) = <i>x</i> has degree 1. We prove that when <i>m</i> = <i>n</i> - 1 (i.e. the point {<i>n</i>} is not in the range), it must be the case that deg(<i>f</i>) = <i>n</i> - <i>o</i>(<i>n</i>). This shows an interesting threshold phenomenon. In fact, the same bound on the degree holds even when the image of the polynomial is any (strict) subset of {0,...,<i>n</i>}. Going back to the case <i>m</i> = <i>n</i>, as we noted the function <i>f</i>(<i>x</i>) = <i>x</i> is possible, however, we show that if one excludes all degree 1 polynomials then it must be the case that deg(<i>f</i>) = <i>n</i> - <i>o</i>(<i>n</i>). Moreover, the same conclusion holds even if <i>m</i> = <i>O</i>(<i>n</i>^1.475--ε). In other words, there are no polynomials of intermediate degrees that map {0,...,<i>n</i>} to {0,...,<i>m</i>}. Furthermore, we give a meaningful answer when <i>m</i> is a large polynomial, or even exponential, in <i>n</i>. Roughly, we show that if <i>m</i> < (^{<i>n</i>/<i>c</i>} _<i>d</i>), for some constant <i>c</i>, then either deg(<i>f</i>) ≤ <i>d</i> - 1 (e.g. <i>f</i>(<i>x</i>) = (^{<i>x</i>-<i>n</i>/2}_{<i>d</i> - 1}) is possible) or deg(<i>f</i>) ≥ <i>n</i>/3 - <i>O</i>(<i>d</i> log <i>n</i>). So, again, no polynomial of intermediate degree exists for such <i>m</i>. We achieve this result by studying a discrete version of the problem of giving a lower bound on the minimal <i>L</i>_∞, norm that a monic polynomial of degree <i>d</i> obtains on the interval [-1,1]. We complement these results by showing that for every <i>d</i> = <i>o</i>(√<i>n</i>/log <i>n</i>) there exists a polynomial <i>f</i>: {0,...,<i>n</i>} → {0,...,<i>O</i>(<i>n</i>^<i>d</i>+0.5)} of degree <i>n</i>/3 - <i>O</i>(<i>d</i> log <i>n</i>) ≤ deg(<i>f</i>) ≤ <i>n</i> - <i>O</i>(<i>d</i> log (<i>n</i>)). Our proofs use a variety of techniques that we believe will find other applications as well. One technique shows how to handle a certain set of diophantine equations by working modulo a well chosen set of primes (i.e. a Boolean cube of primes). Another technique shows how to use lattice theory and Minkowski's theorem to prove the existence of a polynomial with certain properties.