Testing Shift-Equivalence of Polynomials by Deterministic, Probabilistic and Quantum Machines

The polynomials ƒ, g E F[X1,…,Xn] are called shift-equivalent if there exists a shift (α1,…, αn) E Fn such that ƒ(X1 + α1,… ,Xn + αn) = g. In three different cases algorithms which produce the set of all shift-equivalences of ƒ, g in polynomial time are designed. Here 1. (1) in the case of a zero-characteristic field F the designed algorithm is deterministic; 2. (2) in the case of a prime residue field F = Fp and a reduced polynomial ƒ, i.e. degXi(ƒ))</ p − 1, 1 </ i </ n, the algorithm is randomized; 3. (3) in the case of a finite field F = Fq of characteristic 2 the algorithm is quantum; for an arbitrary finite field fFq a quantum machine, which computes the group of all shift-selfequivalences of ƒ, i.e. (β1,…, βn) E Fqn such that ƒ(X1 + β1,…,Xn + βn) = ƒ, is designed.