Pseudogenus: Function Theory & RNA Structures

• Pseudogenus is defined as a refinement of the classical genus, measuring the minimal logarithmic subtractions needed for ordered convergence of meromorphic functions.
• In positive geometry, pseudogenus zero ensures that functions yield canonical forms with only simple poles, a necessary condition for one-dimensional applications in scattering amplitudes.
• In RNA structure classification, pseudogenus guides free-energy penalties to control topological complexity, allowing accurate modeling of structures with pseudoknots.

The term "pseudogenus" has been introduced in contemporary mathematical physics literature to classify meromorphic functions, particularly in the context of one-dimensional positive geometry and its applications in scattering amplitudes. Pseudogenus also finds a formally analogous but conceptually distinct role in the classification of topologically complex biological macromolecule structures, such as RNA secondary structures with pseudoknots. Although the meanings differ across domains, both usages leverage topological and analytic invariants to constrain the admissibility or complexity of structures—whether algebraic, geometric, or biological.

1. Pseudogenus in Positive Geometry and Meromorphic Function Theory

The pseudogenus $\psi$ of a meromorphic function is a refinement of the classical notion of genus $g$ as appearing in Hadamard's factorization theorem. For a function $f(z)$ of finite order, the classical representation

$$f(z)=z^n\,e^{P(z)}\prod_i E_p(z/a_i)\prod_j E_q(z/b_j)^{-1}$$

uses Weierstrass elementary factors $E_k(w)=(1-w)\exp(w+w^2/2+\cdots+w^k/k)$ to ensure convergence. The integer $g = \max(p,q)$ is the minimal number of subtracted terms needed for unconditional convergence (absolute on compacts).

Pseudogenus relaxes the requirement of absolute convergence to ordered convergence by modulus of zeros and poles. Specifically, $\psi$ is the smallest $k\geq0$ such that the product

$$f(z) = z^n\,e^{P(z)} \prod_{i=1}^\infty \left[E_\psi\left(1-\tfrac{z}{a_i}\right)\right]^{k_i}$$

with $P$ of degree at most $g$0, converges uniformly on compacts when ordered by $g$1 (Kim et al., 30 Mar 2026). Thus, $g$2 is the minimal number of logarithmic subtraction terms per factor required to control divergence under this relaxed criterion.

2. Special Case: Pseudogenus Zero and Positive Geometry

The case $g$3—pseudogenus zero—is critical for the classification of canonical forms arising from one-dimensional positive geometries. For such $g$4,

$g$5

and its logarithmic differential

$g$6

contains only simple poles, corresponding to the canonical forms associated with unions and weighted sums of real line segments. The main theorem asserts that pseudogenus zero is both necessary and sufficient for a meromorphic $g$7 to be the exponentiated integral of the canonical form of such a geometry (Kim et al., 30 Mar 2026). All admissible canonical forms in this context appear as $g$8 for $g$9 of pseudogenus zero.

3. Examples and Taxonomy of Pseudogenus

The following table summarizes representative functions and their genus and pseudogenus:

Function	Classical Genus ($f(z)$0)	Pseudogenus ($f(z)$1)
$f(z)$2	2	2
$f(z)$3	1	1
$f(z)$4	1	0
$f(z)$5	1	0
$f(z)$6 (Veneziano beta-function)	1	0

Notably, $f(z)$7 and the Veneziano beta function both have $f(z)$8 but admit non-absolute product factorizations with $f(z)$9 due to special cancellation. These examples establish that $$f(z)=z^n\,e^{P(z)}\prod_i E_p(z/a_i)\prod_j E_q(z/b_j)^{-1}$$0, with precise relationships dependent on the distribution of zeros and poles (Kim et al., 30 Mar 2026).

4. Physical and Geometric Constraints from Pseudogenus

Assigning pseudogenus zero imposes stringent decay on the density of poles and zeros. Specifically, for $$f(z)=z^n\,e^{P(z)}\prod_i E_p(z/a_i)\prod_j E_q(z/b_j)^{-1}$$1, the sum $$f(z)=z^n\,e^{P(z)}\prod_i E_p(z/a_i)\prod_j E_q(z/b_j)^{-1}$$2 must converge. In quantum field theory, if $$f(z)=z^n\,e^{P(z)}\prod_i E_p(z/a_i)\prod_j E_q(z/b_j)^{-1}$$3 label particle masses, the related density $$f(z)=z^n\,e^{P(z)}\prod_i E_p(z/a_i)\prod_j E_q(z/b_j)^{-1}$$4 must satisfy

$$f(z)=z^n\,e^{P(z)}\prod_i E_p(z/a_i)\prod_j E_q(z/b_j)^{-1}$$5

precluding towers of states with excessive density growth, such as the Hagedorn spectrum of string theory unless most states decouple. For Kaluza-Klein modes, only compactifications on fewer than three circles without decoupling are compatible (Kim et al., 30 Mar 2026). In continuum limits, analogous integrability conditions on interval densities apply, ensuring that the Cauchy transform exists in principal value.

5. Pseudogenus in RNA Structure Classification

A formally related but distinct concept appears in computational biology for describing RNA secondary structures, where the genus $$f(z)=z^n\,e^{P(z)}\prod_i E_p(z/a_i)\prod_j E_q(z/b_j)^{-1}$$6 of the pairing graph encodes the minimal number of topological handles needed to embed all base-pairing arcs without crossings. Structures without pseudoknots are planar ($$f(z)=z^n\,e^{P(z)}\prod_i E_p(z/a_i)\prod_j E_q(z/b_j)^{-1}$$7), H-type pseudoknots have $$f(z)=z^n\,e^{P(z)}\prod_i E_p(z/a_i)\prod_j E_q(z/b_j)^{-1}$$8, and more complex intertwined structures have higher genus. The McGenus algorithm employs a “pseudogenus-guided” free-energy model incorporating a global penalty $$f(z)=z^n\,e^{P(z)}\prod_i E_p(z/a_i)\prod_j E_q(z/b_j)^{-1}$$9 for topological complexity, offering tractable search and physical realism in the prediction of structures up to genus two and beyond (Bon et al., 2012). Though not identical in formalism or domain, both mathematical and biological usages of the term reflect a methodology for global complexity penalization in structural inference.

6. Broader Implications and Methodological Roles

The concept of pseudogenus unifies diverse phenomena by quantifying global structural complexity, whether analytic (as in meromorphic functions and their canonical forms) or topological (as in biopolymer folding). In the analytic context, pseudogenus provides necessary and sufficient conditions for canonical forms admitting positive geometry representations, constraining both mathematical and physical function spaces. In stochastic modeling of RNA, pseudogenus-informed penalties prevent spurious overfitting to exceedingly intricate pseudoknot formations. Both settings demonstrate the regulatory power of this invariant in distinguishing physically or biologically plausible objects from unconstrained formal possibilities.