p-Fractals and power series–I. Some 2 variable results

Abstract

u 1 , . . . , u r are in k[[x 1 , . . . , x s ]] with k and deg(u 1 , . . . , u r ) finite. Intending applications to Hilbert-Kunz theory, we code the numbers deg(u a 1 1 , . . . , u a r r ) into a function ϕ u , which empirically satisfies many functional equations related to "magnification by p," where p = char k. p-fractals, introduced here, formalize these ideas.In the first interesting case (r = 3, s = 2), the ϕ u are p-fractals. Our proof uses functions ϕ I attached to ideals I and square-free elements h of A = k [[x, y]]. The finiteness of the set of ideal classes in A/(h) and the existence of "magnification maps" on this set show the ϕ I to be p-fractals.We describe further functional equations coming from a theory of reflection maps on ideal classes, and the paper concludes with examples and open questions.

F -signature of pairs: continuity, p -fractals and minimal log discrepancies

“…This approximation of the derivative of the signature function exhibits the same fractal-like behavior observed before in [MT04].…”

“…We then further deduce, for fixed t, that the F -signature is lower semi-continuous as a function on Spec R when R is regular and a is principal. We also point out the close relationship of the signature function in this setting to the works of Monsky and Teixeira on HilbertKunz multiplicity and p-fractals [MT04,MT06]. Finally, we conclude by showing that the minimal log discrepancy of an arbitrary triple (R, ∆, a t ) is an upper bound for the F -signature.…”

p-Fractals and power series—II.

“…In Section 4 we use these conditions to show that if f (x) and g(y) are strongly rational, then the same is true for f (x) + g(y). We prove analogous results for products and powers; as we have noted, this together with [8,Theorem 1] gives our rationality results.…”

“…For ϕ is also a p-fractal, by [8,Lemma 4.2], and therefore the subspace V of Q Ᏽ spanned by ϕ, ϕ, and all their transforms under the operators T q|b is finite dimensional, and evidently stable under the T q|b . A simple calculation done in the proof of [8,Lemma 4.2] shows that T q|b ψ = T q|q−1−b ψ , for any ψ , so V is also stable under reflection. If M = ᏸ(V ), then M is a finite dimensional subspace of Λ 0 containing ᏸ(ϕ), and Lemma 3.9 shows that S(M) ⊆ p−1 k=0 λ k M. This is essentially the characterization of p-fractals that we were after.…”

“…Let V be a finite dimensional p-stable subspace of Q Ᏽ containing ϕ f and the constant function 1. Let (t) = mt, and define V * as in [8,Lemma 3.8]. Namely, V * consists of all ϕ : Ᏽ → Q whose restriction to each interval [d/m, = q s+s − q s − deg x q , f a q s − deg y q , g a = q s deg x q , f a + q s deg y q , g a − deg x q , f a deg y q , g a , and dividing by q s+s we conclude that ϕ fg = ϕ f + ϕ g − ϕ f ϕ g .…”