# On the degree of univariate polynomials over the integers

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## Abstract

We study the following problem raised by von zur Gathen and Roche [6]:

*What is the minimal degree of a nonconstant polynomial f*: {0,..., *n*} → {0,..., *m*}? Clearly, when *m* = *n* the function *f*(*x*) = *x* has degree 1. We prove that when *m* = *n* — 1 (i.e. the point {*n*} is not in the range), it must be the case that deg(*f*) = *n* — *o*(*n*). 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,..., *n*}. Going back to the case *m* = *n*, as we noted the function *f*(*x*) = *x* is possible, however, we show that if one excludes all degree 1 polynomials then it must be the case that deg(*f*) = *n* — *o*(*n*). Moreover, the same conclusion holds even if *m*=*O*(*n* ^{1.475-ϵ}). In other words, there are no polynomials of intermediate degrees that map {0,...,*n*} to {0,...,*m*}.

Furthermore, we give a meaningful answer when *m* is a large polynomial, or even exponential, in *n*. Roughly, we show that if \(m < (_{\,\,\,d}^{n/c} )\), for some constant *c*, and *d*≤2*n*/15, then either deg(*f*) ≤ *d*—1 (e.g., \(f(x) = (_{\,\,\,d - 1}^{x - n/2} )\) is possible) or deg(*f*) ≥ *n*/3 - *O*(*d*log*n*). So, again, no polynomial of intermediate degree exists for such *m*. We achieve this result by studying a discrete version of the problem of giving a lower bound on the minimal *L* ^{∞} norm that a monic polynomial of degree *d* obtains on the interval [—1,1].

We complement these results by showing that for every integer *k* = *O*(\(\sqrt n \)
) there exists a polynomial *f*: {0,...,*n*}→{0,...,*O*(2^{ k })} of degree *n*/3-*O*(*k*)≤deg(*f*)≤*n*-*k*.

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 a somewhat not too high and not too low degree, for example of degree *n*−*Ω*(logn) for *m*=*n*−1.

## Mathematics Subject Classification (2000)

11C08 12Y05 11D04 68R05 11B83## Preview

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