4

To my knowledge, there exist polynomial-time algorithms for systems of multivariate quadratic equations, i.g., XL(eXtended Linearization). But I don't know if there exists a polynomial-time algorithm for a general system of multivariate cubic equations. Could anybody give an example for me? Thanks very much!

Jianting Wang
  • 43
  • 6
  • 2
    This question appears to be off-topic because it fits mathoverflow.net better – Alma Do Jan 05 '14 at 22:26
  • @AlmaDo Maybe a little. It isn't directly related to programming. But I think 'algorithm' and 'computation complexity' also belong to the topics here. – Jianting Wang Jan 06 '14 at 01:09

2 Answers2

1

Exact solution

Exact solution is analysis solution that can be used to find roots with equations coefficients. I.e. some kind of "formula" to resolve an issue. If that is your problem, then in common case - no way - because of Abel-Ruffini theorem that states about roots for equations of power >=5: such equations can not be resolved in algebraic numerics (i.e. be written with radicals). And you will face such equations even for system from two cubic equations.

Approximate solutions

To do this, you can use one of root finding algorithms, for example, Aberth's method - but you should be aware that it's complexity can not be estimated easily, so if performace is the issue, then look into fast Fourier transform.

Alma Do
  • 37,009
  • 9
  • 76
  • 105
1

XL runs in polynomial time only if the system is overdefined.

In general case, every system of multivariate nonlinear equations over GF(2) is equivalent to some 3-SAT instance. Hence the problem of finding solution is NP-hard.

I can suggest two other methods, which are applicable in general (and in my cases were much faster than XL):

max taldykin
  • 12,459
  • 5
  • 45
  • 64
  • @JiantingWang, don't hesitate to ask me directly if you have any other question on the topic. – max taldykin Jan 06 '14 at 02:20
  • I think I've made a mistake in the description that XL can only be applied to quadratic equations. The similar relationship exists between cubic monomials. Maybe in cubic case its complexity is not polynomial. @max – Jianting Wang Jan 07 '14 at 02:48
  • It is not always polynomial even in quadratic case. Quote from the [abstract](http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.99.6031): "median of the running times of the original XL-algorithm applied to systems of m = n+c quadratic equations in n variables over K which have a solution in K is not subexponential in n". – max taldykin Jan 07 '14 at 03:35