Related papers: Bounds on the Hilbert-Kunz Multiplicity

This paper concerns the question of whether a more direct limit can be used to obtain the limit Hilbert-Kunz multiplicity, a possible candidate for a characteristic zero Hilbert-Kunz multiplicity. The main goal is to establish an…

We give new upper bounds for the number of nonconstant holomorphic maps depending only on the genus. Our estimates improve previously known bounds. The proof is based on the study of pullbacks of holomorphic differentials, together with…

We show that the Hilbert-Kunz multiplicity is a rational number for an R_+-primary homogeneous ideal I=(f_1, ..., f_n) in a two-dimensional graded domain R of finite type over an algebraically closed field of positive characteristic.…

Here we compute Hilbert-Kunz functions of any nontrivial ruled surface over ${\bf P}^1_k$, with respect to all ample line bundles on it.

We present a unified approach to the study of Hilbert-Kunz multiplicity, F-signature, and related limits governed by Frobenius and Cartier linear actions in positive characteristic commutative algebra. We introduce general techniques that…

For a compact spinc manifold $X$ with boundary $b_1(\partial X)=0$, we consider moduli spaces of solutions to the Seiberg-Witten equations in a generalized double Coulomb slice in $L^2_1$ (i.e., $W^{1,2}$) Sobolev regularity. We prove they…

We prove boundary H\"older and Lipschitz regularity for a class of degenerate elliptic, second order, inhomogeneous equations in non-divergence form structured on the left-invariant vector fields of the Heisenberg group. Our focus is on the…

In this article, I give an iterative closed form formula for the Hilbert-Kunz function for any binomial hypersurface in general, over any feild of arbitrary positive characteristic. I prove that the Hilbert-Kunz multiplicity associated to…

We determine the Hilbert-Kunz function of plane elliptic curves in odd characteristic, as well as over arbitrary fields the generalized Hilbert-Kunz functions of nodal cubic curves. Together with results of K. Pardue and P. Monsky, this…

The theory of boundary regularity for $p$-harmonic functions is extended to unbounded open sets in complete metric spaces with a doubling measure supporting a $p$-Poincar\'e inequality, $1<p<\infty$. The barrier classification of regular…

We study Hilbert-Kunz multiplicity of non-singular curves in positive characteristic. We analyse the relationship between the Frobenius semistability of the kernel sheaf associated with the curve and its ample line bundle, and the HK…

We develop new methods to study $\mathfrak{m}$-adic stability in an arbitrary Noetherian local ring. These techniques are used to prove results about the behavior of Hilbert-Samuel and Hilbert-Kunz multiplicities under fine…

This paper is devoted to the proof of Lipschitz regularity, down to the microscopic scale, for solutions of an elliptic system with highly oscillating coefficients, over a highly oscillating Lipschitz boundary. The originality of this…

We develope a local theory for frames on finite dimensional Hilbert spaces. In particular, a bounded frame on a finite dimensional Hilbert space contains a subset which is a good Riesz basis for a percentage (arbitrarily close to one) of…

We prove that, under certain assumptions, generalized Hilbert-Kunz multiplicities can be expressed as linear combinations of classical Hilbert-Kunz multiplicities.

We first present the mixed Hilbert-Samuel multiplicities of analytic local rings over \mathbb{C} as generalized Lelong numbers and further represent them as intersection numbers in the context of modifications. As applications, we give…

In this paper, we give an improved Morse index bound of minimal hypersurfaces from Almgren-Pitts min-max construction in any closed Riemannian manifold $M^{n+1}$ $(n+1 \geq 3$), which generalizes a result by X. Zhou…

We explore the equimultiplicity theory of the $F$-invariants Hilbert--Kunz multiplicity, $F$-signature, Frobenius Betti numbers, and Frobenius Euler characteristic over strongly $F$-regular rings. Techniques introduced in this article…

Upper bounds are given for the weight distribution of binary weakly self-dual codes. To get these new bounds, we introduce a novel method of utilizing unitary operations on Hilbert spaces. This method is motivated by recent progress on…

The main purpose of this paper is to provide formulas for the Hilbert-Kunz multiplicity of fiber product rings and Nagata idealizations. We give explicit formulas for the Hilbert-Kunz multiplicity of a fiber product $R \times_T S$, where…