On Representations of Algebraic Polynomials by Superpositions of Plane Waves

* The author was supported by NSF Grant No. DMS 9706883.

Let P be a bi-variate algebraic polynomial of degree n with the real senior part, and Y = {yj }1,n an n-element collection of pairwise noncolinear unit vectors on the real plane. It is proved that there exists a rigid rotation Y^φ of Y by an angle φ = φ(P, Y ) ∈ [0, π/n] such that P equals the sum of n plane wave polynomials, that propagate in the directions ∈ Y^φ .