Proof of a special case of Shanks´ conjecture

In 1972 Shanks conjectured that the least squares inverse of a two-dimensional polynomial is stable, and verified the conjecture numerically for certain low-degree two-dimensional polynomials. Recently the conjecture was proved false. However, in this note we prove the conjecture for all polynomials of a restricted and low degree. The key to the verification lies in utilizing the centrosymmetric properties of the Toeplitz matrix which arises in an equation yielding the coefficients of the approximate inverse.