A closer look at Witt rings for forms of higher degree

Witt rings for nondegenerate forms φ of degree d over a field of characteristic 0 or greater than d were defined by Harrison and Pareigis. We revisit and discuss their definition as well as some special cases, classify the H-forms employed in their definition, and define Witt rings of diagonal forms of degree d. We also define two new Witt rings for nondegenerate forms φ of degree d.