## 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.

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## References

- [1]
R. C. Baker, G. Harman and J. Pintz: The difference between consecutive primes, II,

*Proceedings of the London Mathematical Society***83**(2001), 532–562. - [2]
R. Beigel: The polynomial method in circuit complexity, in:

*Structure in Complexity Theory Conference*, 82–95, 1993. - [3]
H. Buhrman and R. de Wolf: Complexity measures and decision tree complexity: a survey,

*Theor. Comput. Sci.***288**(2002), 21–43. - [4]
H. Cramér: On the order of magnitude of the difference between consecutive prime numbers,

*Acta Arithmetica***2**(1936), 23–46. - [5]
W. Feller:

*An Introduction to Probability Theory and Its Applications*, volume 1, Wiley, New York, 3rd edition, 1968. - [6]
J. von zur Gathen and J. R. Roche: Polynomials with two values,

*Combinatorica***17**(1997), 345–362. - [7]
O. Goldreich:

*Computational Complexity: A Conceptual Perspective*, Cambridge University Press, 2008. - [8]
P. Gopalan:

*Computing with Polynomials over Composites*, PhD thesis, Georgia Institute of Technology, August 2006. - [9]
P. M. Gruber and C. G. Lekkerkerker:

*Geometry of Numbers*, North-Holland, 1987. - [10]
D. E. Knuth:

*The Art of Computer Programming, Volume III: Sorting and Searching*, Addison-Wesley, 1973. - [11]
M. N. Kolountzakis, R. J. Lipton, E. Markakis, A. Mehta and N. K. Vishnoi: On the Fourier spectrum of symmetric Boolean functions,

*Combinatorica***29**(2009), 363–387. - [12]
N. Linial, Y. Mansour and N. Nisan: Constant depth circuits, Fourier transform and learnability,

*J. ACM***40**(1993), 607–620. - [13]
E. Lucas: Théorie des fonctions numériques simplement périodiques,

*American Journal of Mathematics***1**(1878), 184–196. - [14]
J. C. Mason and D. C. Handscomb:

*Chebyshev Polynomials*, Chapman & Hall/CRC, Boca Raton, FL, 2003. - [15]
E. Mossel, R. O’Donnell and R. A. Servedio: Learning functions of k relevant variables,

*J. Comput. Syst. Sci.***69**(2004), 421–434. - [16]
A. A. Razborov: Lower bounds on the size of bounded depth circuits over a complete basis with logical addition,

*Math. Notes***41**(1987), 333–338. - [17]
H. Robbins: A Remark of Stirling’s Formula,

*American Mathematical Monthly***62**(1955), 26–29. - [18]
A. Shpilka and A. Tal: On the minimal Fourier degree of symmetric Boolean functions,

*Combinatorica***34**(2014), 359–377. - [19]
R. Smolensky: Algebraic methods in the theory of lower bounds for Boolean circuit complexity, in:

*Proceedings of the 19th Annual STOC*, pages 77–82, 1987. - [20]
R. Vein and P. Dale:

*Determinants and Their Applications in Mathematical Physics*, Springer, 1999.

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## Additional information

This research was partially supported by the Israel Science Foundation (grant number 339/10).

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Cohen, G., Shpilka, A. & Tal, A. On the degree of univariate polynomials over the integers.
*Combinatorica* **37, **419–464 (2017). https://doi.org/10.1007/s00493-015-2987-0

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### Mathematics Subject Classification (2000)

- 11C08
- 12Y05
- 11D04
- 68R05
- 11B83