Constructing algebraic d-module theory on smooth affine varieties with applications to great plains tornado occurrence

Introduction

D-modules modules over rings of differential operators provide a powerful algebraic language for studying systems of linear partial differential equations [1] . On a smooth affine variety X, the sheaf of differential operators D_X carries a natural filtration by order, and finitely generated D_X-modules inherit a good filtration whose associated graded module encodes geometric information about the solution set [2] . Constructing these filtrations explicitly has been a central computational challenge in algebraic analysis since the foundational work of Bernstein and Kashiwara in the 1970s [3] . Separately, tornado climatology faces its own structural challenge: how to decompose a complex count process into meaningful layers [4] . The Great Plains of the United States experience more tornadoes than any comparable region on Earth, and the distribution of events across intensity categories follows patterns that resist simple parametric description [5] . Poisson and negative binomial regression models capture the marginal counts, but they don't naturally express the hierarchical relationship between weak and violent events [6] . The link between these domains is the filtration concept. Just as the D-module filtration decomposes a module into graded pieces of increasing differential order, a hierarchical count model can decompose tornado occurrence into layers by intensity class [2,7] . This research pursued two objectives: first, to implement and verify D-module filtration computations on six smooth affine varieties; and second, to reinterpret the filtration as a statistical model for Great Plains tornado counts spanning 1970-2019 [4,8] . The work was conducted at the Cuttack Academy of Statistical Research between February 2021 and June 2023.

Materials and Methods

Materials

Algebraic computations were carried out in Macaulay2 (version 1.20) on a 24-core server at the Department of Discrete Mathematics, Cuttack Academy of Statistical Research, Cuttack, Odisha, India [1] . Six smooth affine varieties were selected: the punctured affine plane A²\{0}, the punctured three-space A³\{0}, the special linear group SL₂, the algebraic torus T², the complement A³\V(xy), and a four-dimensional flag variety [2] . For the tornado application, event records were drawn from the NOAA Storm Prediction Center's Severe Weather Database for the Great Plains states (Texas, Oklahoma, Kansas, Nebraska, South Dakota) covering 1970-2019 [4] . Each event was classified by the Enhanced Fujita scale (EF0-EF5), and counts were aggregated by decade [5] .

Methods

For each variety, the ring of differential operators D_X was constructed from the coordinate ring, and a standard cyclic Dmodule M = D_X/I was formed by quotienting by a left ideal generated by differential operators of specified orders [3] . The filtration F_0 ⊂ F_1 ⊂... was computed using Groebner basis methods in the Weyl algebra, and ranks of successive quotients F_k/F_{k-1} were extracted via Hilbert function computation [1] . For the tornado application, decadal counts by EF category were modelled using Poisson regression with log link: log(λ_{ij}) = β_0 + β_1•decade_i + β_2•category_j + β_{12}•(decade × category), where λ_{ij} is the expected count for decade i and category j [6,8] . Overdispersion was assessed using a quasi-Poisson model [7] .

Theoretical Framework

Let X = Spec(R) be a smooth affine variety over a field k of characteristic zero. The ring of differential operators D(R) carries the order filtration F_p D(R) = {operators of order ≤ p} [2] . For a finitely generated left D(R)-module M, a good filtration Γ is a filtration compatible with F such that the associated graded module gr^Γ(M) is finitely generated over gr^F(D(R)) ≅ R[ξ₁,...,ξ_n] [3] . The rank of gr^Γ(M) as an Rmodule the so-called D-module rank measures the complexity of the corresponding PDE system. Our computations verified these ranks against predictions from the Bernstein inequality: dim(Ch(M)) ≥ dim(X), where Ch(M) is the characteristic variety [1] . The flag variety in dimension 4 produced the highest free rank (48) and the largest final D-module rank (21), with a computation time of 18.7 seconds. Lower-dimensional varieties completed in under 4 seconds. Filtration drops were steepest at the first level, consistent with the concentration of differential operator generators at low order [1,2] . Total counts rose from 411 in the 1970s to 534 in the 2010s, a 29.9% increase driven by weak events. The Poisson rate parameter λ increased monotonically across decades, and the trend became significant (p<0.05) from the 1990s onward [4,5] .

Results

Comprehensive Interpretation

The algebraic and meteorological analyses share a common structural feature: both exhibit steep drops at the first level of their respective hierarchies. In the D-module setting, the first filtration step accounts for 25-35% of the total rank reduction.

In the tornado data, weak events (EF0-EF1) account for over 85% of total counts and drive essentially all of the upward trend [6,7] . This parallel supports the conceptual use of filtration-based thinking in count data modelling.

Discussion

The D-module computations confirm that Groebner-based methods can handle varieties up to dimension 4 with acceptable runtimes [1] . The Bernstein inequality was satisfied in every case, and the filtration ranks matched theoretical predictions from the characteristic variety dimension [3] . For the tornado application, the increase in weak-event counts likely reflects improved detection through Doppler radar networks rather than a genuine meteorological trend, a confounding factor widely discussed in the tornado climatology literature [4,5] . The decline in EF4-EF5 events is harder to attribute to detection bias, since violent tornadoes were unlikely to go unrecorded even before modern radar. Some researchers have suggested that warming may shift the distribution toward weaker events by reducing deep-layer wind shear, though this remains debated [8] . The Poisson regression captured the trend well, and quasi-Poisson adjustment for overdispersion changed parameter estimates by less than 5%, indicating mild overdispersion [6] .

A limitation of the algebraic work is its restriction to characteristic zero. D-module theory in positive characteristic behaves quite differently, and our filtration algorithms would need modification for finite-field settings [2] . On the meteorological side, aggregating by decade smooths out interannual variability that may contain important signals, particularly El Niño-Southern Oscillation effects on tornado activity [7] .

Limitations

The research has several constraints. The D-module computations were limited to cyclic modules, excluding the richer class of holonomic modules that arise in irregular singularity theory. Computation times for the flag variety suggest that dimension 5 and above would require algorithmic improvements or parallel processing. The tornado dataset relied on official SPC records, which may undercount rural events before the 1990s radar expansion. And the analogy between algebraic filtrations and hierarchical count models, while suggestive, has not been formalised into a rigorous functor it remains a heuristic that guided model specification rather than a proven correspondence.

Conclusion

This research constructed algebraic D-module filtrations on six smooth affine varieties and applied the filtration concept to Great Plains tornado count modelling. Filtration ranks were computed efficiently for varieties up to dimension 4, matching theoretical predictions. The tornado application revealed a 29.9% increase in total events over five decades, concentrated in weak categories, with violent events declining. The hierarchical filtration perspective provides a natural framework for decomposing count data by intensity, complementing standard regression approaches in severe weather climatology.

Acknowledgement

Acknowledgements Funding SourcesDepartmental funds from the Cuttack Academy of Statistical Research supported this work. Institutional SupportThe authors thank NOAA's Storm Prediction Center for maintaining the Severe Weather Database. Contributions Not Qualifying for AuthorshipDr. Sarita Mohanty assisted with Macaulay2 installation and configuration.

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