The geometry of higher-order modern portfolio theory

Battle Count: 78

The paper provides important theoretical foundations for higher-order portfolio optimization, which is relevant to quantitative trading in several ways: (1) It characterizes the non-convex landscape that numerical solvers encounter, helping practitioners understand why local minima arise; (2) It provides exact counts of critical portfolios, enabling global optimization strategies; (3) The discriminant locus identifies parameter regimes where portfolio stability changes; (4) The feasible portfolio variety generalizes the efficient frontier to higher-order moments. However, the paper is purely theoretical with no direct trading strategy, backtesting, or empirical implementation, limiting its immediate practical applicability.

Extremely high complexity. The paper requires deep expertise in algebraic geometry (Bézout's theorem, elimination theory, Jacobian analysis, homogenization, discriminant computation), multivariate polynomial systems, and mathematical finance. Practical implementation would require: (1) Computational algebraic geometry software (Macaulay2, Singular); (2) Solving systems of high-degree polynomial equations; (3) Computing discriminant loci via elimination; (4) Handling the rapidly growing degree of feasible portfolio varieties (d·(d-1)·...·(d-n+2)); (5) Translating complex algebraic results to real, positive portfolio weights. The theoretical results are elegant but bridging to practical portfolio management requires substantial additional work.

The paper is purely theoretical with self-contained proofs. All key computations (e.g., Example 3.4, Example 4.2, Example 5.3) are verified using Macaulay2, a publicly available software system. The mathematical framework is fully specified with explicit formulas. However, no code repository is provided, and reproducing the algebraic computations (especially the degree-8 discriminant polynomial with 128 terms) requires significant expertise in computational algebraic geometry.