oppolynomial

An oppolynomial is a polynomial P in a commutative ring R[x] of degree n whose coefficients satisfy a_k = - a_{n-k} for all k. If P(x) = sum_{k=0}^n a_k x^k, this condition is equivalent to x^n P(1/x) = - P(x). In particular, when n is even the middle coefficient must be zero, and when the characteristic of R is 2, the relation becomes a_k = a_{n-k}, so oppolynomials coincide with reciprocal polynomials in that setting.

Notes on characteristic: In characteristic 2, the defining relation a_k = - a_{n-k} reduces to a_k = a_{n-k}, making oppolynomials coincide with reciprocal (palindromic) polynomials in that setting. This illustrates how the base arithmetic influences the symmetry properties of the polynomial.

Examples: For degree 2, taking a_0 = 1, a_2 = -1, a_1 = 0 yields P(x) = 1 - x^2, which satisfies x^2 P(1/x) = - P(x). For degree 3, choosing a_0 = 1, a_3 = -1 and a_1 = 2, a_2 = -2 gives P(x) = 1 + 2x - 2x^2 - x^3, which also satisfies the defining relation.

Relation and use: Oppolynomials are related to reciprocal polynomials through a sign twist and can serve as simple models of anti-symmetric behavior under the transformation x -> 1/x. They appear mainly in theoretical discussions and didactic examples illustrating symmetry properties in polynomials. See also self-reciprocal (palindromic) polynomials and reciprocal polynomials.