An algorithm for associative bilinear forms

Polarization algebras and their relations

“…is obtained by only one elementary transformation from (2). The same result we can get if we consider the automorphism (x − ys, y + zs − xt, z − yt, s + t 2 , t) obtained from (1) by one elementary transformation.…”

“…A counterexample of dimension 9 for m = 3 is given in [28]. [4] started his construction from the automorphism (2). It might be interesting try to get a binary counter-example starting from the binary automorphism (1).…”

“…It has been proved [6,8] that commutative power-associative nil algebras of dimension ≤ 8 are solvable. This problem is also equivalently related to possessing of symmetric associative bilinear forms [2]. Commutative power-associative algebras with a nil basis were studied in [9,21].…”

“…An example of a homogeneous 4-tuple H of degree 3 with nilpotent Jacobian matrix J(H) such that P H X is not nilpotent is given by G. Gorni and G. Zampieri [19]. In fact, the corresponding automorphism (x + s(xt − ys), y + t(xt − ys), s + t 3 , t) is obtained by only one elementary transformation from (2). The same result we can get if we consider the automorphism (x − ys, y + zs − xt, z − yt, s + t 2 , t) obtained from (1) by one elementary transformation.…”

“…A well known quadratic homogeneous automorphism [13, page 98] (x − ys, y + zs − xt, z − yt, s, t) (1) of the affine space A 5 over a field k corresponds to the Suttles' example of a commutative power-associative 5 dimensional algebra which is solvable but not nilpotent [23]. Another well known homogeneous automorphism [13, page 97] (x + s(xt − ys), y + t(xt − ys), s, t) (2) of the affine space A 4 gives an example of a ternary symmetric power-associative 4 dimensional algebra which is also solvable but not nilpotent. Recall that (1) is a tame automorphism and (2) is the first candidate to be non-tame in the case of 4 variables.…”