Polyseries: Formal Power Series Computation

• Polyseries are formal power series beginning at degree one, characterized by truncation and recursive inversion for efficient computation.
• The article demonstrates that Catalan number expansions yield closed-form truncated solutions for quadratic congruences over finite fields.
• It details computational representations using dynamic arrays and pseudocode in languages like Python and C++ to perform formal series operations.

to=arxiv_search 天天中彩票有人말? to=arxiv_search.search code Polyseries, in the Wildberger–Rubine framework, denotes a possibly infinite sequence of coefficients

$\A=\bigl[a_1,a_2,a_3,\dots\bigrangle \quad\longleftrightarrow\quad A(t)=\sum_{k=1}^{\infty} a_k t^k,$

viewed as a formal power series in an indeterminate $t$ over a ring or field $R$. The implementation-oriented study of this framework reviews Peano arithmetic, computable naturals, and non-standard models; defines core data and procedure operations on polyseries; and applies Catalan number expansions to quadratic congruences in a finite field $\mathbb{Z}_p$. Its main result gives a closed form poly series solution in terms of Catalan numbers (Brahimi, 6 Jul 2025).

1. Definition and algebraic structure

A Wildberger polyseries is defined as a formal series beginning at degree $1$,

$\A=\bigl[a_1,a_2,a_3,\dots\bigrangle \quad\longleftrightarrow\quad A(t)=\sum_{k=1}^{\infty} a_k t^k.$

For computation one works with a truncation of order $n$,

$\T_n(\A)=\bigl[a_1,a_2,\dots,a_n\bigrangle \quad\longleftrightarrow\quad A_n(t)=\sum_{k=1}^{n} a_k t^k \quad (\bmod\; t^{n+1}).$

The set $\Polyseries_R$ of formal, possibly truncated, series is endowed with term-wise addition and Cauchy multiplication: $(\A+\B)(t)=\sum_{k\ge 1}(a_k+b_k)t^k, \qquad (\A\cdot \B)(t)=\sum_{n\ge 2}\Bigl(\sum_{i+j=n} a_i b_j\Bigr)t^n.$ In the formulation under discussion, if $t$0 is invertible in $t$1 then $t$2 has a formal inverse $t$3 satisfying $t$4, and its coefficients are obtained by the standard recursive formula for power-series inversion (Brahimi, 6 Jul 2025).

This algebraic presentation places polyseries squarely within formal power-series calculus, but with emphasis on truncation and coefficient-level implementation. The paper’s preliminary review of Peano arithmetic, computable naturals, and non-standard models suggests a foundational interest in how such formal objects are encoded and manipulated computationally, although the concrete development centers on the series operations themselves (Brahimi, 6 Jul 2025).

2. Computational representation

A polyseries can be represented in a high-level language by a dynamic array, or vector, of coefficients. In the illustrative pseudocode, coeffs[0] is unused because the series starts at $t$5, and truncation is implemented by slicing the coefficient array. Addition truncated to length $t$6 fills a fresh array degree by degree, while multiplication uses naïve convolution truncated at order $t$7 (Brahimi, 6 Jul 2025).

The inversion routine is recursive. Writing the inverse coefficients as $t$8, one imposes

$t$9

which yields a recursive computation of $R$0 from previously computed coefficients. Truncation is therefore not an auxiliary convenience but a primary computational mode: all operations preserve the ring structure modulo $R$1 (Brahimi, 6 Jul 2025).

The implementation emphasis is deliberately lightweight. The source states that the code-like descriptions are easily translatable between Python, C++, or similar languages, and the overall machinery can be coded in a few dozen lines of array-manipulating pseudocode (Brahimi, 6 Jul 2025). This gives the polyseries framework a strongly algorithmic character: the core algebra is formal, but its intended use is explicit finite computation with truncations.

3. Catalan expansions and quadratic congruences

A central application is the truncated solution of

$R$2

in $R$3. The series solution is expressed through Catalan numbers: $R$4 To verify the solution up to $R$5, one writes

$R$6

and computes

$R$7

using the Catalan recurrence

$R$8

Hence $R$9 (Brahimi, 6 Jul 2025).

The same method is then applied to the finite-field congruence

$\mathbb{Z}_p$0

After rearranging to $\mathbb{Z}_p$1, setting $\mathbb{Z}_p$2 and $\mathbb{Z}_p$3 produces

$\mathbb{Z}_p$4

so the Catalan-series solution yields

$\mathbb{Z}_p$5

Multiplying by $\mathbb{Z}_p$6 gives the closed form

$\mathbb{Z}_p$7

The source further states that, since $\mathbb{Z}_p$8 splits into two “half-sets” depending on sign, one obtains two distinct solutions in $\mathbb{Z}_p$9 and $1$0 by choosing the corresponding branch of the square-root in the generating function (Brahimi, 6 Jul 2025).

4. Explicit finite-field example

The worked example takes $1$1 and solves

$1$2

in $1$3. The relevant Catalan numbers are

$1$4

The Catalan expansion gives

$1$5

Because the original equation is $1$6, the source then takes the negative of this series, obtaining

$1$7

and gives the complementary solution

$1$8

Both are stated to satisfy $1$9 by direct substitution up to order $\A=\bigl[a_1,a_2,a_3,\dots\bigrangle \quad\longleftrightarrow\quad A(t)=\sum_{k=1}^{\infty} a_k t^k.$0 (Brahimi, 6 Jul 2025).

This example is significant because it exhibits the general method in fully explicit truncated form: the local solution of a quadratic congruence is not merely shown to exist formally, but is written coefficient-by-coefficient in terms of Catalan numbers. The same summary states that representing polynomial equations’ local solutions as formal power series and invoking the universal Catalan-series expansion yields explicit truncated solutions in any sufficiently large prime field $\A=\bigl[a_1,a_2,a_3,\dots\bigrangle \quad\longleftrightarrow\quad A(t)=\sum_{k=1}^{\infty} a_k t^k.$1 (Brahimi, 6 Jul 2025).

5. Terminological distinctions and related series theories

The term “polyseries” is not uniform across the literature summarized here. In the Wildberger–Rubine setting it denotes a formal power series, typically truncated for computation (Brahimi, 6 Jul 2025). In Maple’s MultivariatePowerSeries library, by contrast, “polyseries” is used informally to mean multivariate formal power series and univariate polynomials whose coefficients are such series; the library implements lazy evaluation, homogeneous-part caching, Inverse, Divide, TaylorShift, Weierstrass preparation, and Hensel factorization (Asadi et al., 2021).

A distinct but near-homonymous notion is the Pólya series. In the noncommutative setting, a Pólya series over a field $\A=\bigl[a_1,a_2,a_3,\dots\bigrangle \quad\longleftrightarrow\quad A(t)=\sum_{k=1}^{\infty} a_k t^k.$2 is a rational formal power series $\A=\bigl[a_1,a_2,a_3,\dots\bigrangle \quad\longleftrightarrow\quad A(t)=\sum_{k=1}^{\infty} a_k t^k.$3 whose nonzero coefficients lie in a finitely generated subgroup $\A=\bigl[a_1,a_2,a_3,\dots\bigrangle \quad\longleftrightarrow\quad A(t)=\sum_{k=1}^{\infty} a_k t^k.$4; Bell and Smertnig prove that rational Pólya series are exactly unambiguous rational series and exactly the Hadamard sub-invertible rational series (Bell et al., 2019). In the multivariate commutative setting, a Pólya series is a formal power series $\A=\bigl[a_1,a_2,a_3,\dots\bigrangle \quad\longleftrightarrow\quad A(t)=\sum_{k=1}^{\infty} a_k t^k.$5 with all coefficients in $\A=\bigl[a_1,a_2,a_3,\dots\bigrangle \quad\longleftrightarrow\quad A(t)=\sum_{k=1}^{\infty} a_k t^k.$6, and the structural theorem for $\A=\bigl[a_1,a_2,a_3,\dots\bigrangle \quad\longleftrightarrow\quad A(t)=\sum_{k=1}^{\infty} a_k t^k.$7-finite Pólya series shows equivalence between $\A=\bigl[a_1,a_2,a_3,\dots\bigrangle \quad\longleftrightarrow\quad A(t)=\sum_{k=1}^{\infty} a_k t^k.$8-finiteness, rationality, finite unambiguous decomposition into skew-geometric series, and piecewise multiplicative-exponential behavior on simple linear sets (Bell et al., 2022).

Usage	Core object	Source
Wildberger polyseries	Formal power series $\A=\bigl[a_1,a_2,a_3,\dots\bigrangle \quad\longleftrightarrow\quad A(t)=\sum_{k=1}^{\infty} a_k t^k.$9, often truncated	(Brahimi, 6 Jul 2025)
Informal Maple “polyseries”	Multivariate formal power series and UPoPS	(Asadi et al., 2021)
Pólya series	Rational or $n$0-finite series with coefficients in $n$1	(Bell et al., 2019, Bell et al., 2022)

These usages are mathematically adjacent but not identical. A plausible implication is that “polyseries” functions as a local label for formal-series computation in some settings, whereas “Pólya series” denotes an arithmetic restriction on coefficients in automata-theoretic and $n$2-finite classifications.

6. Scope, extensions, and significance

Within the Wildberger implementation framework, polyseries serve as computable carriers for local solutions of polynomial congruences. The principal example is quadratic: Catalan numbers provide the closed form truncated solution, and the construction is explicit enough to be implemented directly with arrays and recursive coefficient updates (Brahimi, 6 Jul 2025). This makes the framework simultaneously algebraic and procedural.

The summarized account also states that the same machinery extends via Fuss–Catalan numbers to higher-degree equations (Brahimi, 6 Jul 2025). This suggests a broader program in which enumerative generating-function identities become solution formulas for formal congruence problems over finite fields. In that sense, polyseries occupy an intermediate position between symbolic algebra, finite-field computation, and formal power-series methods.

In adjacent research areas, formal series with arithmetic constraints have been classified using very different techniques. Noncommutative Pólya series are characterized via weighted automata, unambiguity, and unit equations (Bell et al., 2019), while multivariate $n$3-finite Pólya series are forced into rational normal forms with monomial denominator factors and skew-geometric decompositions (Bell et al., 2022). Although these are distinct theories, together they show that formal power series with rigid coefficient structure frequently admit unusually explicit descriptions. The Wildberger polyseries construction fits naturally into that broader landscape of explicit series normal forms and computable truncation-based algebra.