Abstract:
Let d≥2 be an integer and let F be a field. A form of degree d over F is a polynomial of homogeneous degree d with coefficients in F. In degree d=2 there is an extensive theory of quadratic forms. We consider forms of degree d>2. The following are among the new results we have proved: 1. A nonsingular form over a field of characteristic zero has nonzero Hessian. This was proved by Harrison in degree d=3. We use some basic algebraic geometry and rational differential forms to give a proof valid in all degrees d≥2. 2. The formal differences of split forms constitute an ideal in the Grothendieck ring of higher degree forms. This generalises a well-known result for quadratic forms to higher degree. 3. In the monoid of equivalence classes of nondegenerate forms of degree d≥3, with the tensor product operation, the submonoid generated by the equivalence classes of the hyperbolic forms is free. This is a small step towards answering a question posed by Harrison. 4. Let the base field have characteristic zero. In every odd degree d≥3 there are no nontrivial families of additive invariants of the forms of degree d. In every even degree d there is a nontrivial family of additive invariants of the forms of degree d. The most familiar example is the family of discriminants of quadratic forms. Our proof involves the symbolic method for representing invariants of the forms of degree d. 5. We give a new proof, in characteristic zero, that a nonsingular form of degree d≥3 has a zero Lie algebra. Our proof involves a certain Schur functor and invokes the basis theorem of Akin, Buchsbaum and Weyman. 6. Let F be a field of characteristic zero and let K:F be a field extension. Let f be a form of degree d≥3 in more than three indeterminates with coefficients in F. Then if f is equivalent over K to a hyperbolic form, f must already be equivalent to a hyperbolic form over F. Compare this with the degree 2 case where, for example, the forms Σ(xk²+yk²) are hyperbolic over C but not over R.
Reference:
Keet, A. 1991. Topics in the algebraic theory of higher degree forms. University of Cape Town.
Bibliography: pages 112-114.