On the Degree of Univariate Polynomials Over the Integers

1. On the Degree of Univariate Polynomials Over the Integers Gil Cohen Weizmann Institute Joint work with Amir Shpilka and Avishay Tal

2. The Question

3. The Question What is the minimal degree of a polynomial of the form ? Mmm… 0

4. The Question What is the minimal degree of anon-constant polynomial of the form ? Mmm…

5. What is it to Us? George Boole Alan Turing

6. Original Motivation The degree of a (Boolean multivariate) polynomial is a natural complexity measure for the function it represents [MP68, NS91, Pat92, GR97] Related to other complexity measures. [NS91] Gave a tight lower bound on the degree of a polynomial of the form (assuming dependency in all variables).

7. Original Motivation [GR97] proved an lower bound for symmetric Boolean multivariate polynomials. Can be thought of as univariate polynomials of the form . [GR97] asked what could be said when the range is ?

8. Later Motivation [ST11] Improved lower bounds on the minimal degree of polynomials implies better algorithms for learning symmetric (Boolean!) juntas. In general, formal derivatives increase the range, and at this point good lower bounds might be useful.

9. Observations Minimal degree . When , the minimal degree is (e.g., ). [GR97] For , minimal degree . By the pigeonhole principle, the minimal degree .

10. What is the Real Behavior? Observations minimal degree [GR97] /2

11. Main Result I Theorem I. the minimal degree is at least .

12. Main Result I minimal degree Threshold /2

13. Main Result II minimal degree Hey, I’m over here! ?

14. Main Result II () Theorem II. or Probably an artifact of the proof I’m listening.. Oh, fine by me! A dichotomous behavior - no intermediate degree. Holds .

15. Upper Bounds [GR97] Asked for upper bounds on the minimal degree of a non-constant polynomial ? Best they constructed: degree . Program search gave as well.

16. Main Result III (Upper Bounds) Theorem III. For there exists an s.t. Hence, by Theorem II Holds .

17. Theorem I Ingredients

18. Ingredients for Theorem I • Newton polynomials • Lucas theorem • The gap between consecutive primes • Linear recurrence relations • Newton polynomials • Lucas theorem • The gap between consecutive primes • Linear recurrence relations

19. Newton Polynomials Do polynomials over the integers have integral coefficients? This is the right intuition, but the wrong basis!

20. Newton Polynomials For every , define The set of polynomials is called the Newton Basisfor degree polynomials.

21. Newton Polynomials For a degree polynomial In our case the ’s are all integers!

22. Newton Polynomials Going back to the example

23. Ingredients for Theorem I • Newton polynomials • Lucas theorem • The gap between consecutive primes • Linear recurrence relations • Newton polynomials • Lucas theorem • The gap between consecutive primes • Linear recurrence relations

24. Lucas Theorem Theorem [Luc78].

25. Lucas Theorem Theorem [Luc1878].

26. Lucas Theorem Theorem [Luc1878]. Let , and be a prime. Denote Then,

27. Ingredients for Theorem I • Newton polynomials • Lucas theorem • The gap between consecutive primes • Linear recurrence relations • Newton polynomials • Lucas theorem • The gap between consecutive primes • Linear recurrence relations

28. The Gap Between Consecutive Primes Let be the -th prime number. What is the asymptotic behavior of ? Conjecture [Cra36]: Theorem [Cra36]: Assuming Riemann Hypothesis Unconditionally [BHP01]:

29. Ingredients for Theorem I • Newton polynomials • Lucas theorem • The gap between consecutive primes • Linear recurrence relations • Newton polynomials • Lucas theorem • The gap between consecutive primes • Linear recurrence relations

30. Linear Recurrence Relations If has degree , then determine . Lemma [GR97]: If has degree , then for all

31. Linear Recurrence Relations A linear recurrence of is a linear combination of shifts of . Of course, . Lemma: The degree of a linear recurrence of with summands .

32. Theorem I Proof of a weaker version

33. Proof of Theorem I (weak version) Theorem. Let be non-constant. Then . Proof. By contradiction. There exists a prime . By [GR97], for

34. Proof of Theorem I (weak version) By [GR97], for

35. Proof of Theorem I (weak version) By [GR97], for By Lucas Theorem, for For , Define the polynomial

36. Proof of Theorem I (weak version) By Lucas Theorem, for Define the polynomial

37. Proof of Theorem I (weak version) By Lucas Theorem, for Define the polynomial Since

38. Proof of Theorem I (weak version)

39. Proof of Theorem I (weak version) If is not a constant, then

40. Proof of Theorem I (weak version) Otherwise, since Hence is linear. As takes integer values and its range is smaller than its domain, must be constant – a contradiction! QED

41. How to get a stronger bound? Modulo a prime is nicer to analyze though we loose information. Natural idea: use many primes! How does one combine all pieces of information from different primes? The set of primes should have some structure.

42. A cube of primes Cube Lemma:cube with primes in .

43. Theorem III proof idea

44. Theorem III – proof idea

45. Theorem III – proof idea Proving the existence of a not too-high-degree polynomial with a given range boils down to proving the existence of a short vector in an appropriate lattice. To avoid trivialities, we prove the existence of such vectors that are linearly independent. By the structure of the lattice this implies that one of them has degree .

46. Open Questions

47. Open Questions • Break that barrier! • For the latter will improve the algorithm for learning symmetric juntas. • What is it with that third in Thm II? • Better upper bounds – if exist.. • Find more applications 

48. Thank You!