Search for: All records

We find a formula, in terms of , and , for the value of the ‐pure threshold for the generic homogeneous polynomial of degree in variables over an algebraically closed field of characteristic . We also show that in every characteristic and for all not divisible by , therealwaysexist reduced polynomials of degree in whose ‐pure threshold is a truncation of the base expansion of at some place; in particular, there always exist reduced polynomials whose ‐pure threshold isstrictlyless than . We provide an example to resolve, negatively, a question proposed by Hernandez, Núñez‐Betancourt, Witt, and Zhang, as to whether a list of necessary restrictions they prove on the ‐pure threshold of reduced forms are “minimal” for . On the other hand, we also provide evidence supporting and refining their ideas, including identifying specific truncations of the base expansion of that are always ‐pure thresholds for reduced forms of degree , and computations that show their conditions suffice (ineverycharacteristic) for degrees up to eight and several other situations.

 

Cubic surfaces in characteristic two are investigated from the point of view of prime characteristic commutative algebra. In particular, we prove that the non-Frobenius split cubic surfaces form a linear subspace of codimension four in the 19-dimensional space of all cubics, and that up to projective equivalence, there are finitely many non-Frobenius split cubic surfaces. We explicitly describe defining equations for each and characterize them as extremal in terms of configurations of lines on them. In particular, a (possibly singular) cubic surface in characteristic two fails to be Frobenius split if and only if no three lines on it form a “triangle”.