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102 N. KRUPNIK AND Y. SPIGEL
Example 6.3. Let n = 2k + 1, k ∈ N. Then the polynomial F(A,B):=(A−B+E) A2k−1+B2k−1 (A−B−E)=F3(A,B)F0(A,B)F4(A,B) of degree n is non-trivial and admissible.
Proof. It is not difficult to check that the above defined polynomials F3, F0, F4 satisfy the conditions (2. 6) - (2.7), and hence, the polynomial F(A,B) is admissible. It is also non-trivial. Indeed, let, for example, A := E11, B := E12, A,B ∈ Mn. Then F(A,B) = 0 ̸= F(B,A).
Examples 6.1 and 6.3 prove the Theorem 1.4. Moreover, the following addi- tional statement follows from Example 6.1:
Proposition 6.4. For any even number n ∈ N there exists a non-trivial ad- missible homogeneous polynomial of degree n.
Conjecture 6.5. Let H(A,B) be an admissible homogeneous polynomial of an odd degree. Then H is trivial.
Let us show (for example) that this conjecture is true for n = 3. Assume that H(A,B) is a non-trivial admissible homogeneous polynomial of degree 3. Then H belongs to the class of all non-trivial admissible polynomials of the third degree , which are not of degree two. On the other hand all such polynomials are described in Theorem 1.3. It follows from (1.4), that up to multiplication by a constant
H(A,B) = A3+B3+ABA+BAB−AB2−A2B−BA2−B2A = H(B,A),
and this contradicts to the condition that H(A,B) is non-trivial.
We proved in Sections 4 and 5 that any non-trivial admissible polynomial of second degree admits a representation (1.3) and any non-trivial admissible polynomial of third degree (which is not of second degree) admits a represen- tation (1.5). The variety of possible factorizations of admissible polynomials of higher degrees is much wider. Below in this section we consider some illustrative
examples (mostly for four-degree polynomials).
We start with the following admissible polynomial of degree n = 2m:
Theorem 6.6. Let F1(A,B) and F2(A,B) be two homogeneous polynomials of the same degree m, and let F (A, B) = F1(A, B)F2(A, B). If
(6.1) F1(A, B)F2(A, B) = F2(B, A)F1(B, A), then there exists a polynomial G(A,B) such that
(6.2) F (A, B) = G(A, B)G(B, A).