, Volume 37, Issue 3, pp 419–464 | Cite as

On the degree of univariate polynomials over the integers

  • Gil Cohen
  • Amir Shpilka
  • Avishay TalEmail author
Original Paper


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) = no(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) = no(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≤2n/15, then either deg(f) ≤ d—1 (e.g., \(f(x) = (_{\,\,\,d - 1}^{x - n/2} )\) is possible) or deg(f) ≥ n/3 - O(dlogn). 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 nΩ(logn) for m=n−1.

Mathematics Subject Classification (2000)

11C08 12Y05 11D04 68R05 11B83 


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  1. [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.MathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    R. Beigel: The polynomial method in circuit complexity, in: Structure in Complexity Theory Conference, 82–95, 1993.Google Scholar
  3. [3]
    H. Buhrman and R. de Wolf: Complexity measures and decision tree complexity: a survey, Theor. Comput. Sci. 288 (2002), 21–43.MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    H. Cramér: On the order of magnitude of the difference between consecutive prime numbers, Acta Arithmetica 2 (1936), 23–46.zbMATHGoogle Scholar
  5. [5]
    W. Feller: An Introduction to Probability Theory and Its Applications, volume 1, Wiley, New York, 3rd edition, 1968.zbMATHGoogle Scholar
  6. [6]
    J. von zur Gathen and J. R. Roche: Polynomials with two values, Combinatorica 17 (1997), 345–362.MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    O. Goldreich: Computational Complexity: A Conceptual Perspective, Cambridge University Press, 2008.CrossRefzbMATHGoogle Scholar
  8. [8]
    P. Gopalan: Computing with Polynomials over Composites, PhD thesis, Georgia Institute of Technology, August 2006.Google Scholar
  9. [9]
    P. M. Gruber and C. G. Lekkerkerker: Geometry of Numbers, North-Holland, 1987.zbMATHGoogle Scholar
  10. [10]
    D. E. Knuth: The Art of Computer Programming, Volume III: Sorting and Searching, Addison-Wesley, 1973.zbMATHGoogle Scholar
  11. [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.MathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    N. Linial, Y. Mansour and N. Nisan: Constant depth circuits, Fourier transform and learnability, J. ACM 40 (1993), 607–620.MathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    E. Lucas: Théorie des fonctions numériques simplement périodiques, American Journal of Mathematics 1 (1878), 184–196.MathSciNetCrossRefzbMATHGoogle Scholar
  14. [14]
    J. C. Mason and D. C. Handscomb: Chebyshev Polynomials, Chapman & Hall/CRC, Boca Raton, FL, 2003.zbMATHGoogle Scholar
  15. [15]
    E. Mossel, R. O’Donnell and R. A. Servedio: Learning functions of k relevant variables, J. Comput. Syst. Sci. 69 (2004), 421–434.MathSciNetCrossRefzbMATHGoogle Scholar
  16. [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.CrossRefzbMATHGoogle Scholar
  17. [17]
    H. Robbins: A Remark of Stirling’s Formula, American Mathematical Monthly 62 (1955), 26–29.MathSciNetCrossRefzbMATHGoogle Scholar
  18. [18]
    A. Shpilka and A. Tal: On the minimal Fourier degree of symmetric Boolean functions, Combinatorica 34 (2014), 359–377.MathSciNetCrossRefzbMATHGoogle Scholar
  19. [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.Google Scholar
  20. [20]
    R. Vein and P. Dale: Determinants and Their Applications in Mathematical Physics, Springer, 1999.zbMATHGoogle Scholar

Copyright information

© János Bolyai Mathematical Society and Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.Department of Computer SciencePrinceton UniversityPrincetonUSA
  2. 2.Blavatnik School of Computer ScienceTel Aviv UniversityTel-AvivIsrael
  3. 3.School of MathematicsInstitute for Advanced StudyPrincetonUSA

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