Two-Level Polynomials

• Two-Level Polynomials are structured expressions with two interacting layers, capturing both numerical values and functional dependencies.
• They appear in higher-order complexity as second-order polynomials with arctic degree calculus and in combinatorial counting with controlled degree and depth parameters.
• Their robust algebraic and compositional properties enable precise classification and analysis in areas like decision-tree complexity, representation theory, and finite-field applications.

Searching arXiv for relevant papers on "Two-Level Polynomials" across the main technical usages of the term. Searching arXiv for: "Two-Level Polynomials" second-order polynomials arXiv, counting with two-level polynomials, level-p complexity two-level majority. “Two-Level Polynomials” is not a single universally fixed notion. In current arXiv usage, the phrase denotes several distinct constructions whose common feature is a stratification by two interacting layers. In higher-order complexity theory, it is used for second-order polynomials $P(N,\Lambda)$, where $N$ is a value variable and $\Lambda$ ranges over monotone functions $\ell:\mathbb N\to\mathbb N$; their degree is no longer a natural number but an arctic first-order polynomial in a symbolic variable $D$ (Lim et al., 2023). In combinatorics, a two-level polynomial is a family $\{f_q\}$ in which, for fixed $q$, $f_q(n)$ is polynomial or quasi-polynomial in $n$, while the degree in $n$ and the leading codegree coefficients vary polynomially with $N$0, with a depth parameter measuring how many leading layers exhibit that behavior (Bogart et al., 7 Jul 2025). Other papers use related “two-level” language for iterated-majority decision-tree complexity, level-two Demazure-character polynomials, level-2 complete permutation polynomials, and a map-level versus level-set-level bilipschitz dichotomy (Jansson et al., 2023, Biswal et al., 2019, Rajagopal et al., 2023, Bodin, 2019).

1. Terminological scope

The phrase has several established meanings, and their separation is mathematically important. In the higher-order-complexity line, “Two-Level Polynomials” are explicitly identified with second-order polynomials: formal expressions built from a numerical variable $N$1, a functional variable $N$2, the operations $N$3 and $N$4, and application $N$5 (Lim et al., 2023). In the combinatorial-counting line, the same phrase denotes an infinite sequence $N$6 whose fixed-$N$7 specialization is polynomial or quasi-polynomial in $N$8, with degree and upper codegree coefficients controlled by polynomial functions of $N$9 up to a prescribed depth (Bogart et al., 7 Jul 2025).

Other usages are narrower and context-specific. In Boolean decision-tree complexity, “two-level” refers to two-level iterated majority, and the relevant polynomials are expected-cost polynomials in the Bernoulli parameter $\Lambda$0 whose lower envelope gives level-$\Lambda$1 complexity (Jansson et al., 2023). In representation theory, the symmetric polynomials $\Lambda$2 interpolate between specialized Macdonald polynomials and graded characters of level two Demazure modules (Biswal et al., 2019). In finite-field theory, a polynomial is “complete to level 2” when $\Lambda$3, $\Lambda$4, and $\Lambda$5 are all permutation polynomials (Rajagopal et al., 2023). In Lipschitz geometry, a “two-level” phenomenon separates bilipschitz equivalence of polynomial maps from bilipschitz equivalence of individual level sets (Bodin, 2019).

A recurrent misconception is to treat these meanings as interchangeable. They are not. The higher-order-complexity and combinatorial-counting theories each provide an explicit formal definition of a “two-level polynomial,” but they formalize different kinds of two-parameter structure.

2. Second-order polynomials in higher-order complexity

In the usage of higher-order computational complexity, a univariate second-order polynomial is a formal expression over $\Lambda$6 with one value variable $\Lambda$7 and one functional variable $\Lambda$8, where $\Lambda$9 ranges over

$\ell:\mathbb N\to\mathbb N$0

Its syntax is

$\ell:\mathbb N\to\mathbb N$1

The semantics is structural: given $\ell:\mathbb N\to\mathbb N$2 and $\ell:\mathbb N\to\mathbb N$3, one interprets $\ell:\mathbb N\to\mathbb N$4 as $\ell:\mathbb N\to\mathbb N$5, $\ell:\mathbb N\to\mathbb N$6 and $\ell:\mathbb N\to\mathbb N$7 as the usual operations on $\ell:\mathbb N\to\mathbb N$8, and $\ell:\mathbb N\to\mathbb N$9 as $D$0. Monotonicity and totality of $D$1 are assumed throughout; there is no subtraction (Lim et al., 2023).

This formalism is motivated by higher-order complexity frameworks, including the use of length functionals $D$2 defined by

$D$3

as in second-order complexity for oracle computations. In that setting, $D$4 abstracts the role played by such monotone size functionals. The “two-level” designation refers to the coexistence of a first-order layer of values and a second-order layer of functions (Lim et al., 2023).

Syntactic equivalence is generated by commutativity and associativity of $D$5 and $D$6, distributivity, and congruence under $D$7, $D$8, and $D$9. This matters because the degree theory is required to be well defined modulo those identifications. A key semantic rigidity theorem states that if $\{f_q\}$0 are pairwise syntactically non-equivalent, then there exists $\{f_q\}$1 such that the values $\{f_q\}$2 are pairwise distinct. This rules out accidental collapse of syntactically distinct second-order polynomials at the semantic level (Lim et al., 2023).

3. Arctic degree, composition laws, and normal forms

The degree of a second-order polynomial is not an integer. It is an arctic first-order polynomial in one indeterminate $\{f_q\}$3, where “arctic” means an expression built using $\{f_q\}$4, $\{f_q\}$5, and $\{f_q\}$6. The inductive definition is

$\{f_q\}$7

The interpretation is structural: sums become maxima, products become sums, and each application of $\{f_q\}$8 multiplies the degree by $\{f_q\}$9. In the univariate case, every arctic polynomial stabilizes to an ordinary polynomial for sufficiently large $q$0; this asymptotic polynomial is denoted $q$1 (Lim et al., 2023).

The degree transforms cleanly under two natural notions of composition. The first, written $q$2, replaces every occurrence of $q$3 in $q$4 by $q$5: $q$6 The second, written $q$7, replaces $q$8 in $q$9 by $f_q(n)$0: $f_q(n)$1 Their degree laws are

$f_q(n)$2

and

$f_q(n)$3

Moreover, the nesting depth of $f_q(n)$4 in $f_q(n)$5 equals the ordinary degree of $f_q(n)$6 (Lim et al., 2023).

A representative example is

$f_q(n)$7

for which

$f_q(n)$8

If

$f_q(n)$9

then $n$0 for $n$1, and hence

$n$2

while

$n$3

This illustrates the difference between substituting into the value slot and substituting into the functional slot (Lim et al., 2023).

The same paper establishes a normal form for second-order polynomials using labelled directed acyclic graphs. Leaves are $n$4 and $n$5; internal nodes are labelled by expressions of the form $n$6, where $n$7 is a multivariate first-order polynomial in the immediate child variables; roots are labelled by first-order polynomials without $n$8. After merging and normalization, syntactically equivalent sets of polynomials yield isomorphic labelled DAGs, and distinct normalized nodes can be made semantically distinct by a suitable choice of $n$9. This gives a canonical structural representative up to isomorphism (Lim et al., 2023).

The framework extends to third-order polynomials $n$0, where $n$1 ranges over monotone operators $n$2. The corresponding degree $n$3 is an arctic second-order polynomial in $n$4, with rules

$n$5

and three composition laws, one for replacing $n$6, one for replacing $n$7, and one for replacing $n$8 (Lim et al., 2023). This suggests a hierarchical degree calculus for higher types.

4. Two-level polynomials in combinatorial counting

In the counting literature, a two-level polynomial is an infinite sequence $n$9 indexed by a parameter $N$00, where each $N$01 is polynomial in $N$02 of degree $N$03, and the leading codegree coefficients are polynomial in $N$04 up to a depth $N$05. Here $N$06 is an eventually nonnegative numerical polynomial, $N$07, and $N$08. Writing

$N$09

with $N$10 for $N$11, the requirement is that there exist polynomials $N$12 such that $N$13 whenever $N$14. The quasi-polynomial version allows periodic dependence on $N$15 via a period function $N$16 and coefficient polynomials $N$17 indexed by residue classes $N$18 (Bogart et al., 7 Jul 2025).

Depth is the central refinement. Infinite depth means that all codegree coefficients agree with polynomial functions in $N$19, including the zero coefficients beyond the degree. Finite depth means only the top portion of the coefficient array behaves polynomially in $N$20. Depth $N$21 is used to accommodate formal manipulations in which no coefficient agreement is required (Bogart et al., 7 Jul 2025).

The algebra of these objects is robust. If $N$22 and $N$23 are two-level quasi-polynomials of degrees $N$24 and $N$25, depths $N$26 and $N$27, and period functions $N$28 and $N$29, then their product is a two-level quasi-polynomial of degree $N$30, depth $N$31, and period $N$32. Scalar multiplication by a polynomial $N$33 preserves depth unless $N$34, in which case the resulting depth is $N$35. If the leading coefficient is constant, powers $N$36 normalize to monic two-level quasi-polynomials. Aligned infinite sums and composition in $N$37 with a fixed polynomial $N$38 also preserve the structure, with explicit formulas for the resulting degree, depth, and periods (Bogart et al., 7 Jul 2025).

A general schema proves two-level polynomiality by organizing a counting problem into “objects,” “cruxes,” and “foundations.” The crux captures the $N$39-dependence; the foundation captures the $N$40-dependence. Conditions S1–S9 control how crux types, foundation isomorphisms, and codegrees interact. The resulting theorem implies that a large class of combinatorial counts is two-level polynomial or two-level quasi-polynomial (Bogart et al., 7 Jul 2025).

The examples are broad. For chromatic polynomials, the path family satisfies

$N$41

with degree $N$42 and

$N$43

for $N$44, giving infinite depth. Cycles satisfy

$N$45

with degree $N$46 and depth $N$47. The grid $N$48 has degree $N$49, leading coefficient $N$50, codegree-1 coefficient $N$51, and codegree-2 coefficient $N$52 (Bogart et al., 7 Jul 2025).

Other examples include the partition numbers $N$53, which are quasi-polynomial in $N$54 of degree $N$55; nonattacking rook counts

$N$56

which form a two-level polynomial of degree $N$57 and infinite depth; labeled nonattacking queens, which form a two-level quasi-polynomial of infinite depth via inside-out polytopes; ordered Sidon sets, with degree $N$58 and leading coefficient $N$59; and Sheffer sequences defined by

$N$60

for which $N$61 is a monic two-level polynomial of degree $N$62 and infinite depth (Bogart et al., 7 Jul 2025).

In this usage, “two-level” refers to a two-parameter asymptotic organization: fixed-$N$63 polynomiality in $N$64, with $N$65-polynomial control over degree and upper codegree data. This is conceptually different from the second-order-polynomial formalism, even though both theories use a two-layer structure.

5. Other mathematically distinct uses of the term

In Boolean decision-tree complexity, the relevant object is not a family $N$66 or a second-order term $N$67, but a set of expected-cost polynomials in the Bernoulli parameter $N$68. For a decision tree $N$69 computing a Boolean function $N$70,

$N$71

and for any fixed tree the recurrence

$N$72

implies polynomial dependence on $N$73. The level-$N$74 complexity is therefore the lower envelope of a finite set of polynomials. For the two-level iterated majority $N$75, the lower envelope collapses to a single polynomial,

$N$76

which strictly improves the earlier conjectured polynomial of Jansson (Jansson et al., 2023).

In representation theory, the polynomials

$N$77

form a two-parameter symmetric-function family indexed by dominant weights. They satisfy

$N$78

and $N$79 is the graded character of a level two Demazure module associated to $N$80. For admissible pairs $N$81, $N$82 is Schur positive (Biswal et al., 2019).

In finite-field theory, “two-level” appears through completeness to level $N$83. A polynomial $N$84 is complete to level $N$85 if

$N$86

are all permutation polynomials. For odd characteristic, this yields an explicit level-2 notion. The families

$N$87

are complete to level $N$88 on $N$89, and hence are 2-complete when $N$90; scaled versions become maximally complete, of level $N$91, on fields with a middle subfield (Rajagopal et al., 2023).

In Lipschitz geometry, the family

$N$92

exhibits a two-tier classification phenomenon: distinct parameters $N$93 give polynomial maps that are not right-bilipschitz equivalent at infinity, yet selected level sets can still be bilipschitz equivalent. Over $N$94, $N$95 and $N$96 are not right-bilipschitz equivalent at infinity when $N$97, but the paper constructs explicit bilipschitz maps sending $N$98 to $N$99 and $\Lambda$00 to $\Lambda$01 (Bodin, 2019).

A neighboring but separate notion is the “level of a pair of polynomials” $\Lambda$02 in characteristic $\Lambda$03, defined as the minimal $\Lambda$04 such that some $\Lambda$05 satisfies

$\Lambda$06

This depends only on the rational function $\Lambda$07 and can be infinite (Boix et al., 2019). Despite the shared vocabulary of levels, this theory is not a definition of “two-level polynomials” in the sense of (Lim et al., 2023) or (Bogart et al., 7 Jul 2025).

6. Applications, limitations, and open directions

The second-order-polynomial formalism is designed for higher-order computational complexity. Its degree calculus supports modular reasoning under substitution, oracle composition, and operator nesting. The paper also notes that if oracle machines have running times bounded by second-order polynomials $\Lambda$08 and $\Lambda$09, then the degrees of the composed bounds transform according to the star and circle laws. This gives a refined classification beyond mere $\Lambda$10-nesting depth (Lim et al., 2023).

Its limitations are explicit. The theory assumes $\Lambda$11 is nondecreasing and total, and it relies on the absence of subtraction. Univariate arctic stabilization yields an exact ordinary polynomial for large $\Lambda$12, but in the multivariate arctic setting stabilization only holds up to constant factors. The paper sketches multivariate generalizations and further orders, including interest in monotone functions $\Lambda$13, but does not establish a full theory there (Lim et al., 2023).

The counting theory is intended for uniform asymptotics in two parameters. It packages a large range of graph-polynomial, partition, chess-placement, inside-out-polytope, Sidon-set, and Sheffer-sequence phenomena into a single framework. Its principal technical device is the control of codegree layers through depth. The limitations are equally specific: depth can drop under addition when degrees differ by a nonconstant amount or when leading coefficients cancel; many quasi-polynomial periods grow with codegree; some families, such as products of paths or cycles, yield only finite depth; and open problems remain for flow polynomials, anti-magic squares, uniform period bounds, and large-$\Lambda$14 coefficient asymptotics (Bogart et al., 7 Jul 2025).

The alternative uses of “two-level” have their own boundaries. The generic library for level-$\Lambda$15 complexity becomes too slow for deeper iterated majority, even though thinning, memoization, and exact polynomial comparison suffice for $\Lambda$16 (Jansson et al., 2023). The higher-level-completeness construction for permutation polynomials reaches the maximal level $\Lambda$17 only when the field has a middle subfield; for fields without such a subfield, the paper guarantees level $\Lambda$18 but does not settle level $\Lambda$19 (Rajagopal et al., 2023). In bilipschitz geometry, map-level moduli and fiberwise equivalence coexist, showing that topological or level-set information alone does not determine bilipschitz classification at infinity (Bodin, 2019).

Across these literatures, the phrase “Two-Level Polynomials” consistently signals a structured interaction between two regimes rather than a single algebraic species. In higher-order complexity the two levels are values and functionals; in combinatorial counting they are the $\Lambda$20-variable polynomial law and the $\Lambda$21-parameterized degree-and-coefficient law; in the other usages they refer to two-level circuit composition, level-two representation-theoretic data, level-2 completeness, or a split between map-level and fiber-level geometry. The unifying theme is not a shared definition, but a shared methodology of encoding one layer of polynomial behavior inside another.