Cohomological invariants for higher degree forms

Let $r>2$ be an integer and let $K$ be a field in which $r!$ is invertible. An $r$-form over $K$ is an equivalence class of regular finite-dimensional $K$-multilinear forms of degree $r$. The operation of direct sums allows the definition of a Witt Grothendieck group of $r$-forms over $K$. It becomes a ring with the multiplication induced by the tensor product of $r$-forms. The properties of the Witt Grothendieck ring of $r$-forms for $r>2$ are quite different from the quadratic case. For example, we have unique sum decomposition, but no diagonalization, and there is an invariant commutative $K$-algebra called the center of an $r$-form, which in unknown to the theory of quadratic forms. We study invariants of $r$-forms. One way to obtain invariants is Galois descent: We obtain a Galois cohomological classification for forms of the Fermat equation, which can also be described as so-called separable forms arising in a natural way from separable extensions of $K$. This description allows the definition of cohomological invariants for separable $r$-forms, one of them being equal to the previously mentioned center invariant. We also obtain two invariants called permanent and determinant, the second being only defined if $r$ is even, both with values in the group $H^1(K,\mu_r)=K^*/K^{*r}$. We find that the determinant can be computed by a formula in the coefficients of the $r$-form, which is a generalization of the Leibniz formula for the quadratic determinant and which has already studied by Cayley and others in the 19th century. This extends the determinant to arbitrary $r$-forms of even degree. Another classical invariant for $r$-forms is the discriminant, a homogenous polynomial in the coefficients of an $r$-forms, which vanishes if and only if the corresponding projective hypersurface is singular. We find that the discriminant of a separable $r$-form is essentially equal to its permanent if $r$ is odd, and to its determinant if $r$ is even. To check if these invariants just depend on the motive of the $r$-form, we analyze a formula computing the zeta function of a separable $r$-form over a finite field and obtain a comparison result relating the zeta function and the discriminant of a separable $r$-form over a finite field. Finally we discuss the consequences of the obtained results with respect to a possible definition of hyperbolic $r$-forms