Modular Reduction for Creative Telescoping

• The paper presents modular reduction techniques that compute telescopers by reducing derivatives or shifts modulo exact terms, thereby streamlining annihilator derivation.
• It employs normal-form computations in finite-dimensional quotient spaces, which minimizes the need for bulky certificates during the creative telescoping process.
• Modular reduction generalizes across rational, hypergeometric, and D-finite cases, offering explicit order bounds and insights into operator factorization.

Modular reduction for creative telescoping is a family of reduction-based techniques for deriving differential or recurrence equations for definite integrals and sums by working in quotients modulo “exact terms.” In the differential case, the basic relation is

$$(P-D_yQ)\cdot f=0, \qquad\text{equivalently}\qquad P\cdot f=D_y\cdot g,$$

while in the summation case one seeks

$L(T)=\Delta_y(G).$

In this literature, “modular reduction” does not primarily mean arithmetic modulo primes. Its closest meaning is reduction modulo $D_y(C(x,y))$, modulo $\Delta_y$-summable terms, modulo annihilating ideals, or modulo images of adjoint operators such as $L^*$. Telescopers arise when repeated parameter derivatives or shifts have reduced remainders that become linearly dependent in a finite-dimensional quotient space (Chen et al., 8 May 2025, Bostan et al., 2018, Brochet et al., 2023).

1. Conceptual framework

Creative telescoping starts from the task of eliminating one variable from a parametrized integral or sum while preserving a linear differential or recurrence relation in the remaining parameters. The standard objects are the telescoper $P$ or $L$, which is free of the variable being integrated or summed out, the certificate $Q$ or $g$, which witnesses exactness, and the annihilating operator, which kills the input before elimination. In the $D$-finite setting these operators live in Ore algebras such as

$L(T)=\Delta_y(G).$0

or multivariate analogues $L(T)=\Delta_y(G).$1, and the annihilating ideal

$L(T)=\Delta_y(G).$2

encodes the ambient quotient structure. $L(T)=\Delta_y(G).$3-finiteness is characterized by the finite-dimensionality of $L(T)=\Delta_y(G).$4 over the coefficient field (Chen et al., 8 May 2025).

The reduction viewpoint replaces certificate search by normal-form computation in an appropriate quotient. One reduces derivatives or shifts of the input modulo exact derivatives, exact differences, annihilating ideals, Gröbner bases, or quotient modules, and then searches for a linear dependence among the resulting remainders. That dependence is the telescoper.

Setting	Reduction target	Representative papers
Rational differential	Modulo exact $L(T)=\Delta_y(G).$5-derivatives $L(T)=\Delta_y(G).$6	(Chen et al., 8 May 2025, Bostan et al., 2013)
Hypergeometric summation	Modulo $L(T)=\Delta_y(G).$7-summable terms via residual forms	(Chen et al., 2015, Huang, 2016, Huang, 23 Feb 2026)
D-finite differential or difference	Modulo images of adjoint operators $L(T)=\Delta_y(G).$8 and quotient modules	(Bostan et al., 2018, Brochet et al., 2023, Giesbrecht et al., 2019)

This organization also explains why reduction-based creative telescoping generalizes across input classes: what changes is the representation of quotient classes, not the basic logic of “reduce, confine, detect dependence.”

2. Rational functions as the prototype

For bivariate rational functions, the basic model is Hermite reduction in the integration variable. In its standard form,

$L(T)=\Delta_y(G).$9

with $D_y(C(x,y))$0 squarefree, $D_y(C(x,y))$1, and $D_y(C(x,y))$2. The remainder is a normal form modulo derivatives, and $D_y(C(x,y))$3 is integrable iff the remainder vanishes. Creative telescoping applies the same reduction not just to $D_y(C(x,y))$4, but to the whole sequence

$D_y(C(x,y))$5

If

$D_y(C(x,y))$6

then

$D_y(C(x,y))$7

so the linear relation among remainders is exactly a telescoper. The lecture notes on creative telescoping emphasize that the remainders $D_y(C(x,y))$8 lie in a finite-dimensional $D_y(C(x,y))$9-vector space because their denominators divide the squarefree part of $\Delta_y$0 and their numerator degrees are bounded. They also state that there exists a telescoper of order at most $\Delta_y$1, and that every telescoper corresponds to a linear relation among the $\Delta_y$2; the smallest such relation gives the minimal-order telescoper (Chen et al., 8 May 2025).

This rational case became the template for later reduction-based methods. In the complexity-driven treatment of proper bivariate rational functions, the same idea is turned into an explicit algorithm: perform Hermite reduction iteratively on $\Delta_y$3, write each remainder as $\Delta_y$4, solve the first linear dependence among the $\Delta_y$5, and recover the telescoper and certificate. That work proves polynomial complexity for minimal telescopers on this input class, gives explicit degree bounds, and already contains ingredients later associated with modularization, including evaluation–interpolation Hermite reduction, “lucky points” at which specialization commutes with reduction, and practical use of “usual modular techniques (probabilistic rank estimate)” (Bostan et al., 2013).

The rational differential case also fixes the main interpretation of modular reduction in this area: one works in

$\Delta_y$6

and Hermite remainders are canonical representatives of those classes.

3. Operator images, adjoints, and quotient modules

The passage from rational functions to $\Delta_y$7-finite objects replaces reduction modulo $\Delta_y$8-exact terms by reduction modulo images of more general operators. A central formulation is the generalized Hermite reduction for an arbitrary linear differential operator

$\Delta_y$9

One seeks a $L^*$0-linear canonical form $L^*$1 on $L^*$2 such that

$L^*$3

and $L^*$4 iff $L^*$5. This is a literal normal form modulo the image of an operator. The construction proceeds through weak local reductions at poles and at infinity, an exceptional space

$L^*$6

and the final canonical form

$L^*$7

The quotient $L^*$8 need not be finite-dimensional globally, but on spaces $L^*$9 of rational functions with prescribed poles, Adolphson’s theorem gives a finite-dimensional quotient. This finite-dimensionality is the structural reason the creative telescoping stage terminates (Bostan et al., 2018).

In the telescoping application, the relevant operator is often the adjoint $P$0 of the minimal annihilating operator $P$1 of a cyclic vector. Lagrange’s identity

$P$2

implies that reduction modulo $P$3 is equivalent to reduction modulo exact derivatives after multiplication by the cyclic vector. The algorithm therefore computes reduced rational factors $P$4 recursively and uses the criterion

$P$5

for the telescoping ideal $P$6. This removes the need to solve a fresh certificate equation at every step (Bostan et al., 2018).

The discrete $P$7-finite summation analogue follows the same pattern. With a cyclic vector $P$8 for the summation shift $P$9, one reduces rational functions $L$0 modulo the image of the adjoint recurrence $L$1. The key equivalence is

$L$2

so canonical forms live in

$L$3

Weak and strong reduction are carried out on pole spaces and on the polynomial part at infinity, and certificates are propagated in compact dag form rather than expanded naively (Brochet et al., 2023).

A related operator-level quotient construction appears in the rational discrete case. There one decomposes the denominator into integer-linear types, passes to operator modules $L$4, and performs left scalar division by $L$5. The quotient and remainder,

$L$6

with remainder $L$7, furnish a canonical test: $L$8 This is reduction modulo exact differences in operator form, with a finite-dimensional remainder space determined by a bounded interval of exponents of $L$9 (Giesbrecht et al., 2019).

4. Hypergeometric, $Q$0-hypergeometric, and mixed terms

In the shift setting, the basic replacement for Hermite reduction is Abramov–Petkovšek reduction. A univariate hypergeometric term is written as

$Q$1

where $Q$2 has shift-reduced $Q$3-shift quotient

$Q$4

with $Q$5 the kernel and $Q$6 the shell. The modified Abramov–Petkovšek reduction improves the classical decomposition by producing

$Q$7

where $Q$8 is a residual form

$Q$9

with $g$0 shift-free and strongly prime with $g$1, $g$2, and

$g$3

Reduction-based creative telescoping then proceeds by reducing $g$4 modulo summable terms and searching for the first linear dependence among the residual forms $g$5; the first such dependence gives a minimal telescoper, and the certificate can be omitted from the search phase (Chen et al., 2015).

For bivariate hypergeometric terms, the main technical obstacle is that residual forms are not literally unique. The 2016 order-bounds paper resolves this by showing that significant denominators are unique up to shift-relatedness, that denominators of the remainders of $g$6 are shift-related to $g$7, and, when telescopers exist, that integer-linearity forces all remainders into a common denominator $g$8. The remainders then lie in a finite-dimensional $g$9-vector space of the form

$D$0

which yields an independent termination proof, minimality by first dependence, and explicit lower and upper bounds on telescoper order (Huang, 2016).

A unified version for hypergeometric and $D$1-hypergeometric terms uses $D$2-standard kernels, a polynomial reduction map

$D$3

and $D$4-remainders

$D$5

where $D$6 has $D$7-normal denominator strongly coprime with $D$8, and $D$9 lies in a fixed complement $L(T)=\Delta_y(G).$00. The key new point is confinement: once the significant denominator is integer-linear, all remainders of the $L(T)=\Delta_y(G).$01-shifts can be chosen with a single bounded denominator $L(T)=\Delta_y(G).$02. The minimal telescoper order is then bounded by

$L(T)=\Delta_y(G).$03

with explicit lower bounds as well. In the ordinary hypergeometric case these bounds coincide with the tight 2016 bounds; in the $L(T)=\Delta_y(G).$04-case they produce new lower bounds and upper bounds that are sometimes better and never worse than the previously known ones (Huang, 23 Feb 2026).

The mixed continuous-discrete case, for integrals of bivariate hypergeometric-hyperexponential terms, combines confinement and Hermite-like reduction. With

$L(T)=\Delta_y(G).$05

confinement writes

$L(T)=\Delta_y(G).$06

with $L(T)=\Delta_y(G).$07. Repeated reduction of the shifted terms $L(T)=\Delta_y(G).$08 then takes place in a $L(T)=\Delta_y(G).$09-dimensional quotient. If $L(T)=\Delta_y(G).$10 has no positive integer residue, the confined representative is canonical, and the first linear dependence among the reduced shifts gives a minimal-order telescoper of order at most $L(T)=\Delta_y(G).$11 (Bostan et al., 2016).

5. Integral bases, algebraic functions, and sequence analogues

A second major line of work constructs reductions from integral bases. For algebraic functions

$L(T)=\Delta_y(G).$12

Trager’s Hermite reduction yields

$L(T)=\Delta_y(G).$13

with a squarefree-denominator remainder. Under a double-root-at-infinity condition, the Hermite remainder vanishes exactly for integrable inputs, so the first linear dependence among the remainders of

$L(T)=\Delta_y(G).$14

produces a minimal telescoper. To obtain a finite-dimensional remainder space directly, one adds polynomial reduction: after splitting the Hermite remainder into a squarefree $L(T)=\Delta_y(G).$15-part and an $L(T)=\Delta_y(G).$16-part, one reduces the latter modulo the image of

$L(T)=\Delta_y(G).$17

for a basis $L(T)=\Delta_y(G).$18 chosen so that $L(T)=\Delta_y(G).$19. The resulting additive decomposition has the form

$L(T)=\Delta_y(G).$20

with $L(T)=\Delta_y(G).$21 and $L(T)=\Delta_y(G).$22 in a finite-dimensional complement $L(T)=\Delta_y(G).$23; this yields existence of telescopers and an order bound given by the dimension of the remainder space (Chen et al., 2016).

For Fuchsian $L(T)=\Delta_y(G).$24-finite functions, the same Trager-style reduction extends from algebraic function fields to modules

$L(T)=\Delta_y(G).$25

with $L(T)=\Delta_y(G).$26 fuchsian. One starts from an integral basis $L(T)=\Delta_y(G).$27 with

$L(T)=\Delta_y(G).$28

where $L(T)=\Delta_y(G).$29 is squarefree. Hermite reduction again removes multiple poles, and polynomial reduction in a basis $L(T)=\Delta_y(G).$30 suitable at infinity yields the additive decomposition

$L(T)=\Delta_y(G).$31

The paper proves that $L(T)=\Delta_y(G).$32 iff $L(T)=\Delta_y(G).$33 is integrable and obtains the telescoper-order bound

$L(T)=\Delta_y(G).$34

where $L(T)=\Delta_y(G).$35 (Chen et al., 2016).

Lazy Hermite reduction for algebraic functions removes the up-front requirement of an integral basis. Starting from a suitable basis $L(T)=\Delta_y(G).$36, it reduces

$L(T)=\Delta_y(G).$37

until $L(T)=\Delta_y(G).$38 has only simple poles, enlarging the $L(T)=\Delta_y(G).$39-module generated by $L(T)=\Delta_y(G).$40 only when the reduction step reveals a missing integral element. On its own this does not solve the integrability problem, but after appending a polynomial reduction step and introducing a second basis $L(T)=\Delta_y(G).$41 suitable at infinity, one again obtains an additive decomposition

$L(T)=\Delta_y(G).$42

with $L(T)=\Delta_y(G).$43 iff $L(T)=\Delta_y(G).$44 is integrable. This supplies the finite-dimensional remainder space needed for creative telescoping in two variables (Chen et al., 2021).

The sequence analogue is reduction-based creative telescoping for P-recursive sequences via integral bases. In a module

$L(T)=\Delta_y(G).$45

one defines suitable bases $L(T)=\Delta_y(G).$46 for finite-place reduction and local integral bases $L(T)=\Delta_y(G).$47 at infinity. The finite-place reduction normalizes denominators to shift-free form, while the infinity reduction bounds degrees, leading to the additive decomposition

$L(T)=\Delta_y(G).$48

In the bivariate setting, one reduces successive parameter shifts $L(T)=\Delta_y(G).$49 to compatible remainders and extracts a telescoper from the first linear dependence over $L(T)=\Delta_y(G).$50 (Chen et al., 2023).

6. Minimality, factorization, and limitations

Reduction-based methods compute telescopers in quotient modules, and this quotient-level minimality has consequences. The residue-based analysis of non-minimality makes the point sharply: a minimal telescoper for an integrand or summand need not be the minimal annihilating operator of the resulting definite integral or sum. In the rational differential case, residues are exactly the obstructions to integrability, and a telescoper annihilates all local residue contributions separately as classes modulo exact derivatives. A definite contour integral, however, may involve only special linear combinations of those residues, and those combinations can satisfy a lower-order relation. The example

$L(T)=\Delta_y(G).$51

illustrates this gap: the integral has minimal annihilator $L(T)=\Delta_y(G).$52, while the integrand has minimal telescoper

$L(T)=\Delta_y(G).$53

The same phenomenon reappears in discrete form through discrete residues and zero-sum submodules (Chen et al., 6 Feb 2025).

This quotient viewpoint also explains factorization. In the submodule approach, creative telescoping is formulated as annihilation of an element $L(T)=\Delta_y(G).$54 in a $L(T)=\Delta_y(G).$55-module $L(T)=\Delta_y(G).$56. If $L(T)=\Delta_y(G).$57 is a nontrivial submodule, then

$L(T)=\Delta_y(G).$58

is a right factor of the minimal telescoper, and the remaining factor is obtained by telescoping $L(T)=\Delta_y(G).$59 inside $L(T)=\Delta_y(G).$60. If $L(T)=\Delta_y(G).$61 decomposes as a direct sum $L(T)=\Delta_y(G).$62, often via automorphisms, then the left factor becomes

$L(T)=\Delta_y(G).$63

This modular reduction by submodules can expose reducible telescopers early and can greatly reduce coefficient swell when the factors are computed separately (Hoeij, 2024).

Practical limitations become acute for multiple sums. The multiple-sum case study on a triple binomial sum emphasizes that certificate management, boundary corrections, singularities at summation boundaries, and noncommutation with moving limits can dominate the runtime. It explicitly states that no creative telescoping implementation currently exists that can resolve all these issues automatically, and it highlights the necessity of the certificate in such computations. From the reduction perspective, this shows that finite-dimensional quotient computations are often only one part of the problem: exact boundary analysis may remain the decisive symbolic layer (Koutschan et al., 2020).

Taken together, these developments define modular reduction for creative telescoping as a quotient-based methodology. Its basic operations are reduction modulo exact derivatives or differences, modulo operator images such as $L(T)=\Delta_y(G).$64, modulo annihilating ideals, and modulo submodules. Its computational core is the construction of canonical or confined remainders in finite-dimensional spaces. Its main strength is that telescopers can be extracted by linear algebra without carrying large certificates throughout the computation. Its main limitation is that quotient-level minimality need not coincide with minimality for the final definite object unless residue cancellations, zero-sum submodules, or boundary conditions are modeled explicitly.