arXiv subjects

Vaibhav Pandey

Publications and source records attributed to Vaibhav Pandey.

At least 19 recordsLinked to original sources

arXiv2026

The variety of nilpotent matrices is $F$-regular

We give an elementary proof that the coordinate ring of the variety of nilpotent matrices is $F$-regular; over an infinite field $K$, this ring also arises as the nullcone for the conjugation action of the general linear group $\textrm{GL}_n(K)$ on the polynomial ring $K[X]$, where $X$ is an $n\times n$ matrix of indeterminates. We prove that the divisor class group of the coordinate ring is the cyclic group $\mathbb{Z}/n\mathbb{Z}$. We then study the case of symmetric nilpotent matrices, where the picture is completely different: the coordinate ring is not normal for $n\geqslant 2$; for $K$ algebraically closed of characteristic other than two, we prove that the coordinate ring is an integral domain precisely when $n$ is odd.

math.AC↗

arXiv2026

Two-loop QCD corrections to $ H \rightarrow b + \bar{b} + g $ at higher powers in the dimensional regulator

We compute the two-loop massless QCD corrections to the amplitude of Higgs boson decay to bottom quark pair and a gluon ($H \rightarrow b + \bar{b} + g$) in the higher powers of the dimensional regularization parameter $\epsilon$. The calculation is performed by projecting the amplitude onto the appropriate Lorentz structures related to the process. We also show the numerical behaviour of the form factors for a few sample phase-space points. These amplitudes are necessary ingredients for computing the three-loop virtual corrections to bottom-quark annihilation to Higgs plus jet production at the hadron collisions.

hep-ph↗

arXiv2026

Associated $ZH$ production in gluon fusion process at NLO+NLL

We present precise results for invariant mass distributions and inclusive cross-sections, of associated $ZH$ production through gluon fusion in QCD. We include threshold logarithms at next-to-leading-logarithmic (NLL) accuracy and match the results to full next-to-leading-order (NLO) results with exact top-quark mass dependence in the virtual amplitudes. At 13.6 TeV energy at LHC, the NLO+NLL cross-section increases the NLO counterpart by about $20\%$. For differential distributions, at 3000 GeV, the uncertainties arising out of the unphysical renormalization and factorization scales is about $20\%$ for NLO; whereas for the resummed results it is around $12\%$. We also combine these results to the Drell-Yan type results at N$^3$LL and present the most precise results in hadron collisions.

hep-ph↗

arXiv2026

VV Resummation To NNLO+NNLL At the LHC

We present the resummed predictions for a vector boson pair production at the LHC. We have performed threshold resummation to next-to-next-to-leading logarithmic (NNLL) accuracy, and then matched them to next-to-next-to-leading order (NNLO) QCD results. After resummation, we observe a reduction in the scale uncertainties arising from unphysical renormalization and factorization scales. We find that the resummed corrections add a few per cent to the fixed order results for both ZZ and WW production.

hep-ph↗

arXiv2026

The arithmetic rank of determinantal nullcones

We compute the arithmetic rank as well as the local/étale cohomological dimension of nullcone ideals arising from the classical actions of the symplectic group, the general linear group, and the orthogonal group. We use these calculations to establish striking vanishing results for local cohomology modules supported at these nullcone ideals; this is achieved via a careful analysis of the critical local cohomology modules. The vanishing theorems that we prove are sharp in various respects.

math.AC↗

arXiv2026

Threshold resummation for $Z$-boson pair production at NNLO+NNLL

The production of a pair of on-shell $Z$-bosons is an important process at the Large Hadron Collider. Owing to its large production cross section at the LHC, this process is very useful for SM precision studies, electroweak symmetry breaking sector as well as to unravel the possible new physics. In this work, we have performed the threshold resummation of the large logarithms that arise in the partonic threshold limit $z \to 1$, up to Next-to-Next-to-Leading Logarithmic (NNLL) accuracy. The presence of the two-loop contributions in the process dependent resummation coefficient $g_0$ makes the numerical computation a non-trivial task. After matching the resummed predictions to the Next-to-Next-to-Leading order (NNLO) fixed order results, we present the invariant mass distribution to NNLO+NNLL accuracy in QCD for the current LHC energies. We find that in the high invariant mass region ($Q=1$ TeV), while the NNLO corrections are as large as $83\%$ with respect to the leading order, the NNLL contribution enhances the cross section by additional few percent, about $4\%$ for $13.6$ TeV LHC. In this invariant mass region, the conventional scale uncertainties in the fixed order results get reduced from $3.4\%$ at NNLO to about $2.6\%$ at NNLO+NNLL, and this reduction is expected to be more for higher $Q$ values.

hep-ph↗

arXiv2026

Threshold resummation for $W$-boson pair production at NNLO+NNLL

We present results for threshold resummation of the invariant mass distribution, for on-shell production of a pair of $W$-bosons at next-to-next-to-leading order + next-to-next-to-leading logarithmic (NNLO+NNLL) accuracy in QCD. Owing to its sensitivity to the self-interactions between gauge bosons, this process is important to investigate at the energies of the Large Hadron Collider (LHC). We achieve this resummation by exploiting the factorization properties of the soft and virtual parts of the partonic cross-section. Our analysis has been carried out for the invariant mass distribution up to $Q$ = 2500 GeV. At this highest $Q$ we find that, for 13.6 TeV LHC, the NNLL resummation enhances the NNLO cross-sections by about $6.3\%$ and reduces the conventional scale uncertainties from 6.8\% at NNLO to 4.1\% at NNLO+NNLL. We also estimate the intrinsic uncertainties due to the non-perturbative parton distribution functions at the highest perturbative order, for both fixed-order and resummed results, to be around 3\% for $Q \sim$ 2000 GeV.

hep-ph↗

arXiv2025

The arithmetic rank of the residual intersections of a complete intersection ideal

The arithmetic rank of an ideal in a polynomial ring over an algebraically closed field is the smallest number of equations needed to define its vanishing locus set-theoretically. We determine the arithmetic rank of the generic $m$-residual intersection of an ideal generated by $n$ indeterminates for all $m\geq n$ and in every characteristic. We further give an explicit description of its set-theoretic generators. Our main result provides a sharp upper bound for the arithmetic rank of any residual intersection of a complete intersection ideal in any Noetherian local ring. In particular, given a complete intersection ideal of height at least two, any of its generic residual intersections -- including its generic link -- fails to be a set-theoretic complete intersection in characteristic zero.

math.AC↗

arXiv2025

Bounds on the plus-pure thresholds of some hypersurfaces in (ramified) regular rings

We study the plus-pure threshold (ppt) of hypersurfaces in mixed characteristic. We show that the ppt limits to the $F$-pure threshold (fpt) as we ramify the base DVR. Additionally, we show that analogs of some positive characteristic extremal singularities cannot attain the same `extremal' ppt values in the unramified setting. We also study equations which have controlled ramification when we adjoin their $p$-th roots as well as equations which admit $p$-th roots modulo $p^2$ (or modulo other values), bounding their ppts. In particular, given a complete unramified regular local ring of mixed characteristic $p>0$, $f^p + p^2 g$ does not define a perfectoid pure singularity for any $f$ and $g$. Finally, we compute bounds on the ppt of hypersurfaces related to elliptic curves. This gives examples where the ppt is neither the corresponding fpt in characteristic $p > 0$ nor the lct in characteristic zero. This also provides examples where $p$ times the ppt is not a jumping number, in stark contrast with the characteristic $p > 0$ picture.

math.AC↗

arXiv2025

Next to Soft Threshold Resummation for $VH$ Production

We study the threshold effects for the associated production of a Higgs boson with a massive vector boson $(V=Z,W)$ in the $q\bar{q} \rightarrow V^\star \rightarrow VH$ process at the LHC. By leveraging the universality of threshold logarithms and employing soft-virtual (SV) and next-to-soft virtual (NSV) resummation techniques, we compute threshold corrections to next-to-next-to-leading logarithmic accuracy. After matching the resummed predictions to the Next-to-Next-to-Leading order (NNLO) fixed order results, we present the invariant mass distribution to NNLO$+\overline{\text{NNLL}}$ accuracy in QCD for the current LHC energies and the total production cross sections. The $VH$ production channel is crucial for studying the couplings of the Higgs boson to the vector bosons $(W,Z)$ and understanding the mechanism of electroweak symmetry breaking. Precision measurements of this process help test the validity of the standard model (SM) and can reveal potential deviations indicating new physics.

hep-ph↗

arXiv2024

On the natural nullcones of the symplectic and general linear groups

Consider a group acting on a polynomial ring S over a field K by degree-preserving K-algebra automorphisms. Several key properties of the invariant ring can be deduced by studying the nullcone of the action, that is, the vanishing locus of all non-constant homogeneous invariant polynomials. These properties include the finite generation of the invariant ring and the purity of its embedding in S. In this article, we study the nullcones arising from the natural actions of the symplectic and general linear groups. For the natural representation of the symplectic group (via copies of the regular representation), the invariant ring is a generic Pfaffian ring. We show that the nullcone of this embedding is F-regular in positive characteristic. Independent of characteristic, we give a complete description of the divisor class group of the nullcone and determine precisely when it is Gorenstein. For the natural representation of the general linear group (via copies of the regular representation and copies of its dual), the invariant ring is a generic determinantal ring. The nullcone of this embedding is typically non-equidimensional; its irreducible components are the varieties of complexes introduced by Buchsbaum and Eisenbud. We show that each of these irreducible components are F-regular in positive characteristic. We also show that the Frobenius splittings of the varieties of complexes may be chosen compatibly so that the nullcone is F-pure.

math.AC↗

arXiv2024

Symbolic powers of the generic linkage of maximal minors

Let $I$ be the ideal generated by the maximal minors of a matrix of indeterminates over a field and let $J$ denote the generic link, i.e., the most general link, of $I$. The generators of the ideal $J$ are not known. We provide an explicit description of the lead terms of the generators of $J$ using Gr\"obner degeneration. Indeed, we construct a degeneration which preserves the entire graded Betti table of $J$ on passing to the initial ideal. We leverage this construction to establish the equality of the symbolic and ordinary powers of $J$. Our analysis of the initial ideal readily yields the Gorenstein property of the associated graded ring of $J$, and, in positive characteristic, the $F$-rationality of the Rees algebra of $J$. Using the technique of $F$-split filtrations, we further obtain the $F$-regularity of the blowup algebras of $J$.

math.AC↗

arXiv2024

$F$-purity and the $F$-pure threshold as invariants of linkage

The generic link of an unmixed radical ideal is a prime ideal. We show that the squarefreeness of the initial ideal and $F$-purity are, however, not preserved along generic links. On the flip side, for several important cases in liaison theory, including generic height three Gorenstein ideals and the maximal minors of a generic matrix, we show that the squarefreeness of the initial ideal, $F$-purity, and the $F$-pure threshold are each preserved along generic links by identifying a property of unmixed ideals which propagates along generic links. We use this property to establish the $F$-regularity of the generic links of such ideals. Finally, we study the $F$-pure threshold of the generic residual intersections of a complete intersection ideal and answer a related question of Kim--Miller--Niu.

math.AC↗

arXiv2024

Linkage and $F$-Regularity of Determinantal Rings

In this paper, we prove that the generic link of a generic determinantal ring defined by maximal minors is strongly $F$-regular. In the process, we strengthen a result of Chardin and Ulrich in the graded setting. They showed that the generic residual intersections of a complete intersection ring with rational singularities again have rational singularities. We show that if the said complete intersection is defined by homogeneous elements and is $F$-rational, then in fact, its generic residual intersections are strongly $F$-regular in positive prime characteristic. Hochster and Huneke showed that determinantal rings are strongly $F$-regular; however, their proof is quite involved. Our techniques allow us to give a new and simple proof of the strong $F$-regularity of determinantal rings defined by maximal minors.

math.AC↗

arXiv2024

Counting geometric branches via the Frobenius map and $F$-nilpotent singularities

We give an explicit formula to count the number of geometric branches of a curve in positive characteristic using the theory of tight closure. This formula readily shows that the property of having a single geometric branch characterizes $F$-nilpotent curves. Further, we show that a reduced, local $F$-nilpotent ring has a single geometric branch; in particular, it is a domain. Finally, we study inequalities of Frobenius test exponents along purely inseparable ring extensions with applications to $F$-nilpotent affine semigroup rings.

math.AC↗

arXiv2023

When are the natural embeddings of classical invariant rings pure?

Consider a reductive linear algebraic group $G$ acting linearly on a polynomial ring $S$ over an infinite field; key examples are the general linear group, the symplectic group, the orthogonal group, and the special linear group, with the classical representations as in Weyl's book: for the general linear group, consider a direct sum of copies of the standard representation and copies of the dual; in the other cases take copies of the standard representation. The invariant rings in the respective cases are determinantal rings, rings defined by Pfaffians of alternating matrices, symmetric determinantal rings, and the Plücker coordinate rings of Grassmannians; these are the classical invariant rings of the title, with $S^G\subseteq S$ being the natural embedding. Over a field of characteristic zero, a reductive group is linearly reductive, and it follows that the invariant ring $S^G$ is a pure subring of $S$, equivalently, $S^G$ is a direct summand of $S$ as an $S^G$-module. Over fields of positive characteristic, reductive groups are typically no longer linearly reductive. We determine, in the positive characteristic case, precisely when the inclusion $S^G\subseteq S$ is pure. It turns out that if $S^G\subseteq S$ is pure, then either the invariant ring $S^G$ is regular, or the group $G$ is linearly reductive.

math.AC↗

arXiv2022

Homological properties of pinched Veronese rings

Pinched Veronese rings are formed by removing an algebra generator from a Veronese subring of a polynomial ring. We study the homological properties of such rings, including the Cohen-Macaulay, Gorenstein, and complete intersection properties. Greco and Martino classified Cohen-Macaulayness of pinched Veronese rings by the maximum entry of the exponent vector of the pinched monomial; we re-prove their results with semigroup methods and correct an omission of a small class of examples of Cohen-Macaulay pinched Veronese rings. When the underlying field is of prime characteristic, we show that pinched Veronese rings exhibit a variety of F-singularities, including F-regular, F-injective, and F-nilpotent. We also compute upper bounds on the Frobenius test exponents of pinched Veronese rings, a computational invariant which controls the Frobenius closure of all parameter ideals simultaneously.

math.AC↗

arXiv2020

Personal Food Model

Food is central to life. Food provides us with energy and foundational building blocks for our body and is also a major source of joy and new experiences. A significant part of the overall economy is related to food. Food science, distribution, processing, and consumption have been addressed by different communities using silos of computational approaches. In this paper, we adopt a person-centric multimedia and multimodal perspective on food computing and show how multimedia and food computing are synergistic and complementary. Enjoying food is a truly multimedia experience involving sight, taste, smell, and even sound, that can be captured using a multimedia food logger. The biological response to food can be captured using multimodal data streams using available wearable devices. Central to this approach is the Personal Food Model. Personal Food Model is the digitized representation of the food-related characteristics of an individual. It is designed to be used in food recommendation systems to provide eating-related recommendations that improve the user's quality of life. To model the food-related characteristics of each person, it is essential to capture their food-related enjoyment using a Preferential Personal Food Model and their biological response to food using their Biological Personal Food Model. Inspired by the power of 3-dimensional color models for visual processing, we introduce a 6-dimensional taste-space for capturing culinary characteristics as well as personal preferences. We use event mining approaches to relate food with other life and biological events to build a predictive model that could also be used effectively in emerging food recommendation systems.

cs.MM↗