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Hong Duc Nguyen

Publications and source records attributed to Hong Duc Nguyen.

13 recordsLinked to original sources
arXiv2026

On the perturbations of Noetherian local domains

We study how the properties of being reduced, integral domain, and normal, behave under small perturbations of the defining equations of a noetherian local ring. It is not hard to show that the property of being a local integral domain (reduced, normal ring) is not stable under small perturbations in general. We prove that perturbation stability holds in the following situations: (1) perturbation of being an integral domain for factorial excellent Henselian local rings; (2) perturbation of normality for excellent local complete intersections containing a field of characteristic zero; and (3) perturbation of reducedness for excellent local complete intersections containing a field of characteristic zero, and for factorial Nagata local rings.

math.AC↗
arXiv2026

Equivariant motivic integration on special formal schemes

In this article, we construct an equivariant version of motivic integration on special formal schemes that generalizes our previous work for algebraic varieties. Pointing out the existence of an equivariant Néron smoothening for a flat generically smooth special formal scheme, we prove a change of variables formula in this integration. Finally, the article introduces the motivic Milnor fiber of a formal power series. It predicts that this quantity is the right one to define the motivic Milnor fiber of a germ of complex analytic functions.

math.AG↗
arXiv2024

Mather-Yau's type theorem for higher Nash blowup algebras

In this paper, we establish a Mather-Yau theorem for higher Nash blowup algebras, demonstrating that the isomorphism type of the local ring of any hypersurface singularity, defined over an arbitrary field, is fully determined by its higher Nash blowup algebras. The classical Mather-Yau theorem (1982) asserts that for isolated complex hypersurface singularities, the isomorphism type of the local ring is determined by the Tjurina algebra. In positive characteristic, this result was extended by considering the higher Tjurina algebras by Greuel and Pham (2017) under the assumptions of an algebraically closed ground field and isolated singularities. Our work begins by proving the stability of higher Nash blowup algebras under contact equivalence in a very general framework. Specifically, we show that the higher Nash blowup algebras of any system of elements in an analytic or geometric ring remain invariant under contact equivalence. For complex hypersurface singularities, this stability was conjectured by Hussain, Ma, Yau, and Zuo, and was recently verified by Le and Yasuda. Finally, the converse is established using a classical result of Samuel (1956).

math.AG↗
arXiv2024

Motivic integration on special rigid varieties and the motivic integral identity conjecture

We prove in this paper the original version of Kontsevich and Soibelman's motivic integral identity conjecture for formal functions by developing a novel framework for equivariant motivic integration on special rigid varieties. This theory is built upon our recent research on equivariant motivic integration within the realm of special formal schemes. The central element of our approach lies in demonstrating that two formal models of a given smooth rigid variety can be dominated by a third formal model. Notably, a similar assertion for quasi-compact rigid varieties was obtained by Bosch, Lütkebohmert, and Raynaud in 1993. Consequently, we establish a concept of motivic volume for a special smooth rigid variety, ensuring independence from the selection of its models. We demonstrate that this motivic volume can be extended to a homomorphism from a certain Grothendieck ring of special smooth rigid varieties to the classical Grothendieck ring of varieties. Moreover, our developed motivic volume exhibits a Fubini-type property, which recovers Nicaise and Payne's motivic Fubini theorem for the tropicalization map.

math.AG↗
arXiv2023

Cohomology of contact loci

We construct a spectral sequence converging to the cohomology with compact support of the m-th contact locus of a complex polynomial. The first page is explicitly described in terms of a log resolution and coincides with the first page of McLean's spectral sequence converging to the Floer cohomology of the m-th iterate of the monodromy, when the polynomial has an isolated singularity. Inspired by this connection, we conjecture that if two germs of holomorphic functions are embedded topologically equivalent, then the Milnor fibers of the their tangent cones are homotopy equivalent.

math.AG↗
arXiv2022

Limits of real bivariate rational functions

Given two nonzero polynomials $f, g \in\mathbb R[x,y]$ and a point $(a, b) \in \mathbb{R}^2,$ we give some necessary and sufficient conditions for the existence of the limit $\displaystyle \lim_{(x, y) \to (a, b)} \frac{f(x, y)}{g(x, y)}.$ We also show that, if the denominator $g$ has an isolated zero at the given point $(a, b),$ then the set of possible limits of $\displaystyle \lim_{(x, y) \to (a, b)} \frac{f(x, y)}{g(x, y)}$ is a closed interval in $\overline{\mathbb{R}}$ and can be explicitly determined. As an application, we propose an effective algorithm to verify the existence of the limit and compute the limit (if it exists). Our approach is geometric and is based on Puiseux expansions.

math.CA↗
arXiv2016

Euler reflexion formulas for motivic multiple zeta functions

We introduce a new notion of $\boxast$-product of two integrable series with coefficients in distinct Grothendieck rings of algebraic varieties, preserving the integrability and commuting with the limit of rational series. In the same context, we define a motivic multiple zeta function with respect to an ordered family of regular functions, which is integrable and connects closely to Denef-Loeser's motivic zeta functions. We also show that the $\boxast$-product is associative in the class of motivic multiple zeta functions. Furthermore, a version of the Euler reflexion formula for motivic zeta functions is nicely formulated to deal with the $\boxast$-product and motivic multiple zeta functions, and it is proved using the theory of arc spaces. As an application, taking the limit for the motivic Euler reflexion formula we recover the well known motivic Thom-Sebastiani theorem.

math.AG↗
arXiv2016

The right classification of univariate power series in positive characteristic

While the classification of univariate power series up to coordinate change is trivial in characteristic 0, this classification is very different in positive characteristic. In this note we give a complete classification of univariate power series $f\in K[[x]]$, where $K$ is an algebraically closed field of characteristic $p>0$ by explicit normal forms. We show that the right determinacy of $f$ is completely determined by its support. Moreover we prove that the right modality of $f$ is equal to the integer part of $μ/p$, where $μ$ is the Milnor number of $f$. As a consequence we prove in this case that the modality is equal to the proper modality, which is the dimension of the $μ$-constant stratum in an algebraic representative of the semiuniversal deformation with trivial section.

math.AG↗
arXiv2016

Invariants of plane curve singularities and Plücker formulas in positive characteristic

We study classical invariants for plane curve singularities $f\in K[[x,y]]$, $K$ an algebraically closed field of characteristic $p\geq 0$: Milnor number, delta invariant, kappa invariant and multiplicity. It is known, in characteristic zero, that $μ(f)=2δ(f)-r(f)+1$ and that $κ(f)=2δ(f)-r(f)+\mathrm{mt}(f)$. For arbitrary characteristic, Deligne prove that there is always the inequality $μ(f)\geq 2δ(f)-r(f)+1$ by showing that $μ(f)-\left( 2δ(f)-r(f)+1\right)$ measures the wild vanishing cycles. By introducing new invariants $γ,\tildeγ$, we prove in this note that $κ(f)\geq γ(f)+\mathrm{mt}(f)-1\geq 2δ(f)-r(f)+\mathrm{mt}(f)$ with equalities if and only if the characteristic $p$ does not divide the multiplicity of any branch of $f$. As an application we show that if $p$ is "big" for $f$ (in fact $p > κ(f)$), then $f$ has no wild vanishing cycle. Moreover we obtain some Plücker formulas for projective plane curves in positive characteristic.

math.AG↗
arXiv2015

Right unimodal and bimodal singularities in positive characteristic

The problem of classification of real and complex singularities was initiated by Arnol'd in the sixties who classified simple, unimodal and bimodal w.r.t. right equivalence. The classification of right simple singularities in positive characteristic was achieved by Greuel and the author in 2014. In the present paper we classify right unimodal and bimodal singularities in positive characteristic by giving explicit normal forms. Moreover we completely determine all possible adjacencies of simple, unimodal and bimodal singularities. As an application we prove that, for singularities of right modality at most 2, the $μ$-constant stratum is smooth and its dimension is equal to the right modality. In contrast to the complex analytic case, there are, for any positive characteristic, only finitely many 1-dimensional (resp. 2-dimensional) families of right class of unimodal (resp. bimodal) singularities. We show that for fixed characteristic $p>0$ of the ground field, the Milnor number of $f$ satisfies $μ(f)\leq 4p$, if the right modality of $f$ is at most 2.

math.AG↗