GRADE: Diagonal and Hadamard Grades

• GRADE is a framework of metrics and models that classify functions by their complexity through diagonal and Hadamard grades, establishing a strict hierarchy from rational to transcendental functions.
• It uses nilpotent monodromy to set sharp lower bounds for grading, linking algebraic structure with analytic properties in D-finite and hypergeometric functions.
• Applications include the analysis of hypergeometric series and Apéry’s generating function, providing concrete examples of grade computations and refined stratifications within function spaces.

GRADE refers to a diverse set of frameworks, metrics, and models across multiple domains including algebra, information retrieval, dialogue evaluation, automated grading, benchmarking, and machine learning. The concept typically encodes a notion of "gradation" or "degree"—either as a measure of complexity, capability, or evaluation along structural, algebraic, or computational axes. This article provides an integrated technical overview of GRADE as it appears in algebraic function theory, especially through the notion of diagonal grade and its relation to nilpotent monodromy, hypergeometric functions, Hadamard grade, and notable generating functions (Harder et al., 14 Apr 2025).

1. Diagonal Grade: Definition and Function Classes

Let $K$ be a number field and consider the formal power series $$h(x_0,\dots,x_n) = \sum_{i_0,\dots,i_n\geq 0} a_{i_0,\dots,i_n}x_0^{i_0}\cdots x_n^{i_n}$$—a rational or algebraic function with $Q(0)\neq 0$. The diagonal of $h$ is the single-variable series: $\Delta_n(h) = \sum_{i \geq 0} a_{i,\dots,i} x^i.$ The diagonal grade $dg(f)$ of a nonzero $f(x)\in K\llbracket x\rrbracket$ is: $$dg(f) = \min\{ n \geq 0 \mid \exists\, h \in K(x_0,\dots,x_n) : f = \Delta_n(h) \}$$ with $dg(f) = \infty$ if no such $n$ exists. Define $$h(x_0,\dots,x_n) = \sum_{i_0,\dots,i_n\geq 0} a_{i_0,\dots,i_n}x_0^{i_0}\cdots x_n^{i_n}$$0 and $$h(x_0,\dots,x_n) = \sum_{i_0,\dots,i_n\geq 0} a_{i_0,\dots,i_n}x_0^{i_0}\cdots x_n^{i_n}$$1. Notably, $$h(x_0,\dots,x_n) = \sum_{i_0,\dots,i_n\geq 0} a_{i_0,\dots,i_n}x_0^{i_0}\cdots x_n^{i_n}$$2 is the set of rational functions and $$h(x_0,\dots,x_n) = \sum_{i_0,\dots,i_n\geq 0} a_{i_0,\dots,i_n}x_0^{i_0}\cdots x_n^{i_n}$$3 is the set of algebraic functions. The diagonal grade thus encodes a hierarchy of function spaces: $$h(x_0,\dots,x_n) = \sum_{i_0,\dots,i_n\geq 0} a_{i_0,\dots,i_n}x_0^{i_0}\cdots x_n^{i_n}$$4 This structure formalizes how increasing the number of variables in the rational function "source" allows for greater function-theoretic richness in the resulting univariate series.

2. Nilpotent Monodromy and Grade Lower Bounds

Consider a $$h(x_0,\dots,x_n) = \sum_{i_0,\dots,i_n\geq 0} a_{i_0,\dots,i_n}x_0^{i_0}\cdots x_n^{i_n}$$5-finite function $$h(x_0,\dots,x_n) = \sum_{i_0,\dots,i_n\geq 0} a_{i_0,\dots,i_n}x_0^{i_0}\cdots x_n^{i_n}$$6 with minimal annihilating operator over $$h(x_0,\dots,x_n) = \sum_{i_0,\dots,i_n\geq 0} a_{i_0,\dots,i_n}x_0^{i_0}\cdots x_n^{i_n}$$7: $$h(x_0,\dots,x_n) = \sum_{i_0,\dots,i_n\geq 0} a_{i_0,\dots,i_n}x_0^{i_0}\cdots x_n^{i_n}$$8 Let $$h(x_0,\dots,x_n) = \sum_{i_0,\dots,i_n\geq 0} a_{i_0,\dots,i_n}x_0^{i_0}\cdots x_n^{i_n}$$9 denote the associated differential module (rank $Q(0)\neq 0$0). The local monodromy at $Q(0)\neq 0$1 decomposes as $Q(0)\neq 0$2, with $Q(0)\neq 0$3 unipotent. The nilpotence index $Q(0)\neq 0$4 is the minimal $Q(0)\neq 0$5 with $Q(0)\neq 0$6.

The principal result in this direction states: $Q(0)\neq 0$7

$Q(0)\neq 0$8

Consequently, for any $Q(0)\neq 0$9-finite $h$0,

$h$1

Thus, the nilpotence of the monodromy yields a fundamental lower bound for the diagonal grade and, by implication, for any related gradation such as the Hadamard grade.

3. Diagonal and Hadamard Grade of Hypergeometric Series

For the generalized hypergeometric function: $h$2 the diagonal and Hadamard grade can be determined via monodromy. In the non-resonant case with $h$3, Levelt's monodromy theorem guarantees the nilpotence index at $h$4 is $h$5. Thus,

$h$6

On the other hand, the explicit Hadamard factorization yields: $h$7 where each $h$8 is algebraic. This gives an upper bound $h$9 for the Hadamard grade. For the special case with $\Delta_n(h) = \sum_{i \geq 0} a_{i,\dots,i} x^i.$0,

$\Delta_n(h) = \sum_{i \geq 0} a_{i,\dots,i} x^i.$1

for all $\Delta_n(h) = \sum_{i \geq 0} a_{i,\dots,i} x^i.$2. This provides strictly increasing sequences of function classes: $\Delta_n(h) = \sum_{i \geq 0} a_{i,\dots,i} x^i.$3 and, for Hadamard grade, likewise for the corresponding $\Delta_n(h) = \sum_{i \geq 0} a_{i,\dots,i} x^i.$4.

4. Hadamard Grade: Definition and Relation

The Hadamard product of $\Delta_n(h) = \sum_{i \geq 0} a_{i,\dots,i} x^i.$5 and $\Delta_n(h) = \sum_{i \geq 0} a_{i,\dots,i} x^i.$6 is: $\Delta_n(h) = \sum_{i \geq 0} a_{i,\dots,i} x^i.$7 The Hadamard grade $\Delta_n(h) = \sum_{i \geq 0} a_{i,\dots,i} x^i.$8 is the minimal $\Delta_n(h) = \sum_{i \geq 0} a_{i,\dots,i} x^i.$9 (or $dg(f)$0) such that

$dg(f)$1

with all $dg(f)$2 algebraic ($dg(f)$3 if $dg(f)$4 is rational). Any Hadamard product of diagonals is again a diagonal, so $dg(f)$5. The correspondence between nilpotence and Hadamard (or diagonal) grade is precise for the hypergeometric family considered above, and the direct Hadamard decomposition gives $dg(f)$6 for

$dg(f)$7

5. Apéry's Generating Function and Higher Grade

Apéry’s sequence: $dg(f)$8 admits the diagonal representation: $dg(f)$9 implying $f(x)\in K\llbracket x\rrbracket$0. Furthermore, $f(x)\in K\llbracket x\rrbracket$1 satisfies Apéry’s third-order Fuchsian ODE with monodromy at $f(x)\in K\llbracket x\rrbracket$2 a $f(x)\in K\llbracket x\rrbracket$3 Jordan block, implying $f(x)\in K\llbracket x\rrbracket$4 and thus $f(x)\in K\llbracket x\rrbracket$5. This confirms the existence of $f(x)\in K\llbracket x\rrbracket$6 with diagonal grade $f(x)\in K\llbracket x\rrbracket$7, resolving an outstanding question on the strictness of the class inclusions $f(x)\in K\llbracket x\rrbracket$8.

6. Structural and Theoretical Implications

The findings substantiate that:

• The diagonal grade provides a strict stratification of $f(x)\in K\llbracket x\rrbracket$9-finite functions, with rational $$dg(f) = \min\{ n \geq 0 \mid \exists\, h \in K(x_0,\dots,x_n) : f = \Delta_n(h) \}$$0 algebraic $$dg(f) = \min\{ n \geq 0 \mid \exists\, h \in K(x_0,\dots,x_n) : f = \Delta_n(h) \}$$1 strictly higher diagonal classes.
• Nilpotent monodromy is a sharp lower bound for both diagonal and Hadamard grade, and for classical hypergeometric series of the form $$dg(f) = \min\{ n \geq 0 \mid \exists\, h \in K(x_0,\dots,x_n) : f = \Delta_n(h) \}$$2, both grades equal $$dg(f) = \min\{ n \geq 0 \mid \exists\, h \in K(x_0,\dots,x_n) : f = \Delta_n(h) \}$$3.
• Hadamard grade, while potentially distinct in general, coincides with diagonal grade in these explicit cases due to direct factorization into algebraic components.
• The explicit computation for Apéry's $$dg(f) = \min\{ n \geq 0 \mid \exists\, h \in K(x_0,\dots,x_n) : f = \Delta_n(h) \}$$4 demonstrates grade $$dg(f) = \min\{ n \geq 0 \mid \exists\, h \in K(x_0,\dots,x_n) : f = \Delta_n(h) \}$$5, answering in the affirmative the existence question for higher grades.

7. Outlook and Extensions

The presented framework establishes deep connections between the algebraic-combinatorial representation of functions, the monodromy theory of their differential equations, and concrete realizations via diagonals and Hadamard products. These techniques are foundational for the classification and understanding of transcendental numbers, the analytic properties of special functions, and applications in periods, enumerative geometry, and mathematical physics (Harder et al., 14 Apr 2025). The extension of grade computations to wider classes of special functions, generating functions in enumerative combinatorics, and their analogues in $$dg(f) = \min\{ n \geq 0 \mid \exists\, h \in K(x_0,\dots,x_n) : f = \Delta_n(h) \}$$6-series or multivariate settings remains a stimulating area for future research.