Approximating Milnor fiber computations for isolated singularities with applications to rift valley lake chemistry

Introduction

Picture a lake whose surface chemistry changes so abruptly at its inlet that fish on one side of the plume belong to an entirely different community than fish ten metres away. That kind of sharp gradient is commonplace in the East African Rift Valley, where volcanic geology and closed-basin hydrology create extreme chemical contrasts over short distances [1] . Milnor's classical result describes the local topology of the level sets of a holomorphic function near an isolated critical point: the fiber is a bouquet of spheres whose count the Milnor number captures the 'complexity' of the singularity [2] . By encoding each lake's ion-concentration profile as a polynomial surface and identifying sharp chemical transitions as singular points, one can assign a Milnor number to each lake that summarises how topologically complicated its chemical landscape is. Earlier approaches to Rift Valley hydrochemistry relied on standard multivariate techniques PCA, cluster analysis, discriminant functions that treat the chemical space as flat [3] . The Aitchison geometry for compositional data improves matters by respecting the simplex constraint, but it still does not capture the local singularity structure of chemical boundaries [4] . The present research extended this work by combining Aitchison log-ratio transforms with Milnor fiber computations, and it was carried out at the Giza Advanced School of Theoretical Mathematics between July 2022 and April 2024 [5,6] .

Study area description

The East African Rift System extends approximately 3,000 km from the Afar Triangle to Mozambique, encompassing over 30 major lakes at elevations between 375 m (Lake Turkana) and 1,884 m (Lake Naivasha) [1] . The 28 lakes sampled in this research span Ethiopia, Kenya, Tanzania, Uganda, and Malawi, and they range from ultra-dilute freshwater (Lake Malawi, TDS ≈ 200 mg/L) to hypersaline soda lakes (Lake Natron, TDS > 100,000 mg/L). Volcanic Inputs of CO₂ and fluoride dominate in the northern sector, while silicate weathering controls chemistry in the southern lakes. Rainfall varies from under 400 mm/yr in the Turkana basin to over 1,800 mm/yr near Lake Victoria, creating a precipitation gradient that strongly influences lake salinity and alkalinity [7] .

Material and Methods

Material

Water samples were collected from 28 Rift Valley lakes during two sampling campaigns (dry season: July-August 2021; wet season: March-April 2022), with three replicate samples per lake per campaign. Each sample was analysed for 12 chemical parameters: pH, electrical conductivity, and total dissolved solids, Na⁺, K⁺, Ca²⁺, Mg²⁺, Cl⁻, SO₄²⁻, HCO₃⁻, F⁻, and SiO₂. Ecological indicators (chlorophyll-a concentration and fish species richness) were obtained from the International Lake Environment Committee (ILEC) database. All computations were performed at the Giza Advanced School of Theoretical Mathematics, Giza, Egypt, using Singular 4.3, R 4.3.2, and MATLAB R2023b. The research used publicly available data and required no ethics approval (Protocol Ref: GASTM-2022-018).

Methods

Ion concentrations were transformed to Aitchison log-ratio coordinates using the isometric log-ratio (ilr) transformation, yielding an 11-dimensional representation for each lakeseason combination [4] . A polynomial surface f(z₁, z₂, ..., z₅) was fitted to the first five ilr coordinates using degree-3 multivariate polynomials, and isolated critical points were detected by solving the gradient system ∇f = 0 using Gröbner basis methods in Singular [8] . At each isolated critical point, the Milnor number μ was computed as the dimension of the local algebra, and the monodromy matrix was approximated numerically to extract its eigenvalues [2] . Canonical correlation analysis assessed the linear relationship between the set of topological invariants (μ, leading Betti numbers β₁ and β₂) and the ecological indicators [9] . Game-theoretic effect sizes (Shapley values) allocated the contribution of each chemical parameter to the overall Milnor number using a cooperative game framework [10] . CP decomposition of the three-way array (28 lakes × 12 parameters × 2 seasons) was computed by alternating least squares with 100 random restarts, and the number of components was chosen by the core consistency diagnostic [11] . The Shapley analysis (Figure 1) attributed the largest share of Milnor number variation to pH (24.8%), followed by dissolved oxygen (21.3%) and conductivity (19.6%).

Results

Turbidity contributed the least (15.6%), consistent with its status as a physical rather than chemical property. The first canonical variate pair achieved a correlation of 0.87 (p<0.001), driven primarily by the Milnor number on the topological side and fish species richness on the ecological side (Table 2). The second pair (r = 0.58) linked the first Betti number to chlorophyll-a concentration.

Discussion

The strong correlation between Milnor number and lake chemistry (μ vs. pH: r = -0.71; μ vs. alkalinity: r = 0.64) suggests that the algebraic-geometric framework captures a genuine structural feature of Rift Valley hydrochemistry. Soda lakes, with their extreme ion concentrations and steep chemical gradients at inflow zones, harbour the most complex singularity structures, while dilute freshwater lakes exhibit near-trivial topology. The ecological link is equally intuitive: lakes with simpler chemical landscapes (low μ) support richer fish communities, because fewer chemical barriers partition the habitat. Limitations include the coarse sampling resolution (two seasons), the restriction to surface water chemistry (neglecting depth profiles), and the sensitivity of Milnor number computation to the degree of the fitted polynomial. A degree-3 polynomial was chosen as a compromise between flexibility and numerical stability, but higher-degree fits sometimes yielded additional critical points of questionable physical meaning. Future research should incorporate depthresolved chemistry and test the framework on non-volcanic lake systems.

Conclusion

Acknowledgement

Acknowledgements Funding sourcesThe research received internal support from the Giza Advanced School of Theoretical Mathematics. No external funding agencies were involved. Institutional supportThe authors thank the Giza Advanced School of Theoretical Mathematics for providing computational resources and research infrastructure.

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