Query-efficient algorithms for polynomial interpolation over composites

@article{Gopalan2006QueryefficientAF,
  title={Query-efficient algorithms for polynomial interpolation over composites},
  author={Parikshit Gopalan},
  journal={SIAM J. Comput.},
  year={2006},
  volume={38},
  pages={1033-1057}
}
  • P. Gopalan
  • Published 22 January 2006
  • Computer Science
  • SIAM J. Comput.
The problem of polynomial interpolation is to reconstruct a polynomial based on its valuations on a set of inputs <i>I</i>. We consider the problem over <i>Z</i><inf>m</inf> when <i>m</i> is composite. We ask the question: Given I ⊆ <i>Z</i><inf>m</inf>, how many evaluations of a polynomial at points in I are required to compute its value at every point in I? Surprisingly for composite <i>m</i>, this number can vary exponentially between log[<i>I</i>] and [<i>I</i>] in contrast to the prime… 
Polynomial approximations over Z / p k Z
We study approximation of Boolean functions by low-degree polynomials over the ring Z/pkZ. More precisely, given a Boolean function F : {0, 1}n → {0, 1}, define its k-lift to be Fk : {0, 1}n → {0,
Computing with polynomials over composites
TLDR
This thesis addresses some such prime vs. composite problems from algorithms, complexity and combinatorics, and the surprising connections between them, and shows that symmetric polynomials can viewed as simultaneous communication protocols.
Constructing Ramsey graphs from Boolean function representations
  • P. Gopalan
  • Mathematics, Computer Science
    21st Annual IEEE Conference on Computational Complexity (CCC'06)
  • 2006
TLDR
The barrier to better Ramsey constructions through such algebraic methods appears to be the construction of lower degree representations, and it is shown that better bounds cannot be obtained using symmetric polynomials.
On polynomial approximations over Z/2kZ
TLDR
It is observed that the model the authors study subsumes the model of non-classical polynomials in the sense that proving bounds in the model implies bounds on the agreement ofNon- classical poynomials with Boolean functions.
Towards understanding the approximation of Boolean functions by nonclassical polynomials
TLDR
The ability of nonclassical polynomials to approximate Boolean functions with respect to both previously studied and new notions of approximation is investigated.
Polynomial Interpolation over the Residue Rings Zn
We consider the problem of polynomial interpolation over the residue rings Zn. The general case can easily be reduced to the case of n = pk due to the Chinese reminder theorem. In contrast to the

References

SHOWING 1-10 OF 51 REFERENCES
Learning Matrix Functions over Rings
TLDR
This paper generalizes the learning algorithm for automata over fields given in [BBB+] with a polynomial query complexity for the class of functions representable as f(x) = Πi=1n Ai(xi), where, for each 1 ≤ i ≤ n, Ai is a mapping Ai : X → Rmi× mi+1 and m1 = mn+1 = 1 .
Dimension reduction for ultrametrics
TLDR
It is proved that an ultrametric on n points can be embedded in l<sup>d</sup><inf>p</inf></i> with distortion at most 1 + ε, and d = O(ε-2 log log n) (this bound matches the best known bound for the special case of an equilateral space).
Lower bounds for approximations by low degree polynomials over Z/sub m/
  • N. Alon, R. Beigel
  • Computer Science
    Proceedings 16th Annual IEEE Conference on Computational Complexity
  • 2001
TLDR
A Ramsey-theoretic argument is used to obtain the first lower bounds for approximations over Z/sub m/ by nonlinear polynomials by suggesting nonapproximability results imply the first known lower bounds on the top fanin of MAJoMOD/ sub m/oAND/sub O(1)/ circuits.
Primality and identity testing via Chinese remaindering
TLDR
These algorithms use fewer random bits and work for a larger class of polynomials than all the previously known methods, e.g. the Schwartz-Zippel test, the Chen-Kao (1997) test and the Lewin-Vadhan test.
Probabilistic approximation of metric spaces and its algorithmic applications
  • Y. Bartal
  • Computer Science, Mathematics
    Proceedings of 37th Conference on Foundations of Computer Science
  • 1996
TLDR
It is proved that any metric space can be probabilistically-approximated by hierarchically well-separated trees (HST) with a polylogarithmic distortion.
Polynomial functions on finite commutative rings
‡ Every function on a finite residue class ring D/I of a Dedekind domain D is induced by an integer-valued polynomial on D that preserves congruences mod I if and only if I is a power of a prime
Representing Boolean functions as polynomials modulo composite numbers
TLDR
The unexpected result that the MOD m -degree of the OR ofN variables is O(\sqrt[\tau ]{N})\), wherer is the number of distinct prime factors ofm, which is optimal in the case of representation by symmetric polynomials.
Constructing Ramsey graphs from Boolean function representations
  • P. Gopalan
  • Mathematics, Computer Science
    21st Annual IEEE Conference on Computational Complexity (CCC'06)
  • 2006
TLDR
The barrier to better Ramsey constructions through such algebraic methods appears to be the construction of lower degree representations, and it is shown that better bounds cannot be obtained using symmetric polynomials.
Symmetric polynomials over Zm and simultaneous communication protocols
...
...