ON THE DEGREE OF RESTRICTIONS OF q-VALUED LOGIC FUNCTIONS TO LINEAR MANIFOLDS

In case of a finite field F-q, the degree of restricting a q-valued logic function in n variables to a r-dimensional linear manifold of the vector space F-q(n) is defined as the degree of a polynomial in r variables that represents this restriction. For manifolds of a fixed dimension, the probability of occurrence of restrictions with a degree not higher than the given one is estimated, and the asymptotics of the number of manifolds on which the restrictions are affine is obtained. It is shown that if n -> 1, for almost all q-valued logic functions in n variables, the value of the maximum dimension of a linear manifold on which the restriction is affine belongs to the segment [left perpendicular log(q) n + + log(q) log(q) n right perpendicular, inverted right perpendicular log(q) n + log(q) log(q) n inverted left perpendicular], while the analogous parameter for the case of fixing variables is in the range [left perpendicular log(q) n right perpendicular, inverted right perpendicular log(q) n inverted left perpendicular].