This paper studies the integration problem in differential fields that may involve quantities reminiscent of the classical Weierstrass ℘ function, which are defined by a first-order nonlinear differential equation. We extend the classical notion of special polynomials to elements of Weierstrass-like extensions and present algorithms for reduction in such extensions. As an application of these results, we derive some new formulae for integrals of powers of ℘.

Elaborating on an approach recently proposed by Mark van Hoeij, we continue to investigate why creative telescoping occasionally fails to find the minimal-order annihilating operator of a given definite sum or integral. We offer an explanation based on the consideration of residues.

A considerable portion of the work on special functions in computer algebra during the past decades was focused on D-finite functions. This focus was chosen for good reasons, as the concept of D-finiteness has proven to provide a fairly good compromise between, on the one hand, covering as many functions as possible, and on the other hand, keeping the class of functions restricted enough that computations stay reasonably efficient. In the talk, we will illustrate how questions about D-finite functions naturally arise in applications and how computer algebra is nowadays routinely used to answer such questions.

We provide bounds on the size of polynomial differential equations obtained by executing closure properties for D-algebraic functions. While it is easy to obtain bounds on the order of these equations, it requires some more work to derive bounds on their degree. Here we give bounds that apply under some technical condition about the defining differential equations.

We propose a way to split a given bivariate P-recursive sequence into a summable part and a non-summable part in such a way that the non-summable part is minimal in some sense. This decomposition gives rise to a new reduction-based creative telescoping algorithm based on the concept of integral bases.

For a given ideal <?TeX $I\subseteq \mathbb {K}[x_1,\dots,x_n,y_1,\dots,y_m]$?> Math 1 in a polynomial ring with n + m variables, we want to find all elements that can be written as f − g for some <?TeX $f\in \mathbb {K}[x_1,\dots,x_n]$?> Math 2 and some <?TeX $g\in \mathbb {K}[y_1,\dots,y_m]$?> Math 3 , i.e., all elements of I that contain no term involving at the same time one of the x1, …, xn and one of the y1, …, ym. For principal ideals and for ideals of dimension zero, we give a algorithms that compute all these polynomials in a finite number of steps.

We propose investigating a summation analog of the paradigm for parallel integration. We make some first steps towards an indefinite summation method applicable to summands that rationally depend on the summation index and a P-recursive sequence and its shifts. There is a distinction between so-called normal and so-called special polynomials. Under the assumption that the corresponding difference field has no unnatural constants, we are able to predict the normal polynomials appearing in the denominator of a potential closed form. We can also handle the numerator. Our method is incomplete so far as we cannot predict the special polynomials appearing in the denominator. However, we do have some structural results about special polynomials for the setting under consideration.

Automated reasoning techniques based on computer algebra have seen renewed interest in recent years and are for example heavily used in formal verification of arithmetic circuits. However, the verification process might contain errors. Generating and checking proof certificates is important to increase the trust in automated reasoning tools. For algebraic reasoning, two proof systems, Nullstellensatz and polynomial calculus, are available and are well-known in proof complexity. A Nullstellensatz proof captures whether a polynomial can be represented as a linear combination of a given set of polynomials by providing the co-factors of the linear combination. Proofs in polynomial calculus dynamically capture that a polynomial can be derived from a given set of polynomials using algebraic ideal theory. In this article we present the practical algebraic calculus as an instantiation of the polynomial calculus that can be checked efficiently. We further modify the practical algebraic calculus and gain LPAC (practical algebraic calculus + linear combinations) that includes linear combinations. In this way we are not only able to represent both Nullstellensatz and polynomial calculus proofs, but we are also able to blend both proof formats. Furthermore, we introduce extension rules to simulate essential rewriting techniques required in practice. For efficiency we also make use of indices for existing polynomials and include deletion rules too. We demonstrate the different proof formats on the use case of arithmetic circuit verification and discuss how these proofs can be produced as a by-product in formal verification. We present the proof checkers Pacheck, Pastèque, and Nuss-Checker. Pacheck checks proofs in practical algebraic calculus more efficiently than Pastèque, but the latter is formally verified using the proof assistant Isabelle/HOL. The tool Nuss-Checker is used to check proofs in the Nullstellensatz format. Supplementary Information: The online version contains supplementary material available at 10.1007/s10703-022-00391-x.

Trager’s Hermite reduction solves the integration problem for algebraic functions via integral bases. A generalization of this algorithm to D-finite functions has so far been limited to the Fuchsian case. In the present paper, we remove this restriction and propose a reduction algorithm based on integral bases that is applicable to arbitrary D-finite functions.

Although in theory we can decide whether a given D-finite function is transcendental, transcendence proofs remain a challenge in practice. Typically, transcendence is certified by checking certain incomplete sufficient conditions. In this paper we propose an additional such condition which catches some cases on which other tests fail.

We introduce a new method for discovering matrix multiplication schemes based on random walks in a certain graph, which we call the flip graph. Using this method, we were able to reduce the number of multiplications for the matrix formats (4,4,5) and (5,5,5), both in characteristic two and for arbitrary ground fields.

A sequence is called C-finite if it satisfies a linear recurrence with constant coefficients. We study sequences which satisfy a linear recurrence with C-finite coefficients. Recently, it was shown that such C2-finite sequences satisfy similar closure properties as C-finite sequences. In particular, they form a difference ring.

Abstract Given a lattice $$L\subseteq \mathbb Z^m$$ L ⊆ Z m and a subset $$A\subseteq \mathbb R^m$$ A ⊆ R m , we say that a point in A is lonely if it is not equivalent modulo $$L$$ L to another point of A. We are interested in identifying lonely points for specific choices of $$L$$ L when A is a dilated standard simplex, and in conditions on $$L$$ L which ensure that the number of lonely points is unbounded as the simplex dilation goes to infinity.

It is known for linear operators with polynomial coefficients annihilating a given D-finite function that there is a trade-off between order and degree. Raising the order may give room for lowering the degree. The relationship between order and degree is typically described by a hyperbola known as the order-degree curve. In this paper, we add the height into the picture, i.e., a measure for the size of the coefficients in the polynomial coefficients. For certain situations, we derive relationships between order, degree, and height that can be viewed as order-degree-height surfaces.

Reconstructing a hypothetical recurrence equation from the first terms of an infinite sequence is a classical and well-known technique in experimental mathematics. We propose a variation of this technique which can succeed with fewer input terms.

Bronstein's lazy Hermite reduction is a symbolic integration technique that reduces algebraic functions to integrands with only simple poles without the prior computation of an integral basis. We sharpen the lazy Hermite reduction by combining it with the polynomial reduction to solve the decomposition problem of algebraic functions. The sharpened reduction is then used to design a reduction-based telescoping algorithm for algebraic functions in two variables.

Currently the most efficient automatic approach to verify gate-level multipliers combines SAT solving and computer algebra. In order to increase confidence in the verification, proof certificates are generated. However, due to different solving techniques, these certificates require two different proof formats, namely DRUP and PAC. A combined proof has so far been missing. Correctness of this approach can thus only be trusted up to the correctness of compositional reasoning. In this paper we show how to generate a single proof in one proof format, which then allows to certify correctness using one simple proof checker. We further investigate empirically the effect on proof generation and checking time as well as on proof size. It turns out that PAC proofs are much more compact and faster to check.

We present an algorithm which for any given ideal I ⊆ K[x, y] finds all elements of I that have the form f(x) - g(y), i.e., all elements in which no monomial is a multiple of xy.

In an earlier paper, the notion of integrality known for algebraic number fields and fields of algebraic functions has been extended to D-finite functions. The aim of the present paper is to extend the notion to the case of P-recursive sequences. In order to do so, we formulate a general algorithm for finding all integral elements for valued vector spaces and then show that this algorithm includes not only the algebraic and the D-finite cases but also covers the case of P-recursive sequences.

Verifying arithmetic circuits and most prominently multiplier circuits is an important problem which in practice still requires substantial manual effort. The currently most effective approach uses polynomial reasoning over pseudo boolean polynomials. In this approach a word-level specification is reduced by a Gröbner basis which is implied by the gate-level representation of the circuit. This reduction returns zero if and only if the circuit is correct. We give a rigorous formalization of this approach including soundness and completeness arguments. Furthermore we present a novel incremental column-wise technique to verify gate-level multipliers. This approach is further improved by extracting full- and half-adder constraints in the circuit which allows to rewrite and reduce the Gröbner basis. We also present a new technical theorem which allows to rewrite local parts of the Gröbner basis. Optimizing the Gröbner basis reduces computation time substantially. In addition we extend these algebraic techniques to verify the equivalence of bit-level multipliers without using a word-level specification. Our experiments show that regular multipliers can be verified efficiently by using off-the-shelf computer algebra tools, while more complex and optimized multipliers require more sophisticated techniques. We discuss in detail our complete verification approach including all optimizations.

We combine SAT and computer algebra to substantially improve the most effective approach for automatically verifying integer multipliers. In our approach complex final stage adders are detected and replaced by simple adders. These simplified multipliers are verified by computer algebra techniques and correctness of the replacement step by SAT solvers. Our new dedicated reduction engine relies on a Gröbner basis theory for coefficient rings which in contrast to previous work no longer are required to be fields. Modular reasoning allows us to verify not only large unsigned and signed multipliers much more efficiently but also truncated multipliers. We are further able to generate and check proofs an order of magnitude faster than in our previous work, relative to verification time, while other competing approaches do not provide certificates.

The currently most effective approach for verifying gate-level multipliers uses Computer Algebra. It reduces a word-level multiplier specification by a Grobner basis derived from a gate-level implementation. This reduction produces zero if and only if the circuit is a multiplier. We improve this approach by extracting full- and half-adder constraints to reduce the Grobner basis, which speeds up computation substantially. Refactoring the specification in terms of partial products instead of inputs yields further improvements. As a third contribution we extend these algebraic techniques to verify the equivalence of bit-level multipliers without using a word-level specification.

Verifying arithmetic circuits, and most prominently multipliers, is an important problem but in practice still requires substantial manual effort. Recent work tries to solve this issue using techniques from computer algebra. The most effective approach uses polynomial reasoning over pseudo boolean polynomials. In this paper we give a rigorous formalization of this approach and present a new column-wise verification technique for the correctness of gate-level multipliers which does not require the reduction of a full word-level specification. We formally prove soundness and completeness of our technique, making use of our precise formalization. Our experiments show that simple multipliers can be verified efficiently by using off-the-shelf computer algebra tools, while more complex and optimized multipliers require more sophisticated techniques. Further, our paper independently confirms the effectiveness of previous related work. We make all benchmarks and tools publicly available.

It is well known that the composition of a D-finite function with an algebraic function is again D-finite. We give the first estimates for the orders and the degrees of annihilating operators for the compositions. We find that the analysis of removable singularities leads to an order-degree curve which is much more accurate than the order-degree curve obtained from the usual linear algebra reasoning.

Continuing a series of articles in the past few years on creative telescoping using reductions, we develop a new algorithm to construct minimal telescopers for algebraic functions. This algorithm is based on Trager's Hermite reduction and on polynomial reduction, which was originally designed for hyperexponential functions and extended to the algebraic case in this paper.

For a nonnegative integer $t$, let $c_t$ be the asymptotic density of natural numbers $n$ for which $s(n+t)\geq s(n)$, where $s(n)$ denotes the sum of digits of $n$ in base $2$. We prove that $c_t>1/2$ for $t$ in a set of asymptotic density $1$, thus giving a partial solution to a conjecture of Cusick stating that $c_t > 1/2$ for all $t$. Interestingly, this problem has several equivalent formulations, for example that the polynomial $X(X+1)\cdots (X+t-1)$ has less than $2^t$ zeros modulo $2^{t+1}$. The proof of the main result is based on Chebyshev's inequality and the asymptotic analysis of a trivariate rational function using methods from analytic combinatorics.

The Abramov-Petkovsek reduction computes an additive decomposition of a hypergeometric term,which extends the functionality of the Gosper algorithm for indefinite hypergeometric summation. We modify the Abramov-Petkovsek reduction so as to decompose a hypergeometric term as the sum of a summable term and a non-summable one. The outputs of the Abramov-Petkovsek reduction and our modified version share the same required properties. The modified reduction does not solve any auxiliary linear difference equation explicitly. It is also more efficient than the original reduction according to computational experiments. Based on this reduction, we design a new algorithm to compute minimal telescopers for bivariate hypergeometric terms. The new algorithm can avoid the costly computation of certificates.

We propose a differential analog of the notion of integral closure of algebraic function fields. We present an algorithm for computing the integral closure of the algebra defined by a linear differential operator. Our algorithm is a direct analog of van Hoeij's algorithm for computing integral bases of algebraic function fields.

The Apagodu-Zeilberger algorithm can be used for computing annihilating operators for definite sums over hypergeometric terms, or for definite integrals over hyperexponential functions. In this paper, we propose a generalization of this algorithm which is applicable to arbitrary δ-finite functions. In analogy to the hypergeometric case, we introduce the notion of proper δ-finite functions. We show that the algorithm always succeeds for these functions, and we give a tight a priori bound for the order of the output operator.

We provide bounds on the size of operators obtained by algorithms for executing D-finite closure properties. For operators of small order, we give bounds on the degree and on the height (bit-size). For higher order operators, we give degree bounds that are parameterized with respect to the order and reflect the phenomenon that higher order operators may have lower degrees (order-degree curves).

Our main result is a formulation and proof of the reverse hypercontractive inequality in the sum-of-squares (SOS) proof system. As a consequence we show that for any constant 0 < γ ≤ 1/4, the SOS/Lasserre SDP hierarchy at degree certifies the statement “the maximum independent set in the Frankl–Rödl graph has fractional size o(1)”. Here is the graph with V = {0,1}n and (x,y) ∊ E whenever Δ(x, y) = (1 – γ)n (an even integer). In particular, we show the degree-4 SOS algorithm certifies the chromatic number lower bound “ ”, even though is the canonical integrality gap instance for which standard SDP relaxations cannot even certify “ ”. Finally, we also give an SOS proof of (a generalization of) the sharp (2, q)-hypercontractive inequality for any even integer q.

Desingularization is the problem of finding a left multiple of a given Ore operator in which some factor of the leading coefficient of the original operator is removed. An order-degree curve for a given Ore operator is a curve in the (r,d)-plane such that for all points (r,d) above this curve, there exists a left multiple of order r and degree d of the given operator. We give a new proof of a desingularization result by Abramov and van Hoeij for the shift case, and show how desingularization implies order-degree curves which are extremely accurate in examples.

We present a new algorithm for computing hyperexponential solutions of linear ordinary differential equations with polynomial coefficients. The algorithm relies on interpreting formal series solutions at the singular points as analytic functions and evaluating them numerically at some common ordinary point. The numerical data is used to determine a small number of combinations of the formal series that may give rise to hyperexponential solutions.

Creative telescoping applied to a bivariate proper hypergeometric term produces linear recurrence operators with polynomial coefficients, called telescopers. We provide bounds for the degrees of the polynomials appearing in these operators. Our bounds are expressed as curves in the (r, d)-plane which assign to every order r a bound on the degree d of the telescopers. These curves are hyperbolas, which reflect the phenomenon that higher order telescopers tend to have lower degree, and vice versa.

We show that the problem of constructing telescopers for rational functions of m + 1 variables is equivalent to the problem of constructing telescopers for algebraic functions of m variables and we present a new algorithm to construct telescopers for algebraic functions of two variables. These considerations are based on analyzing the residues of the input. According to experiments, the resulting algorithm for rational functions of three variables is faster than known algorithms, at least in some examples of combinatorial interest. The algorithm for algebraic functions implies a new bound on the order of the telescopers.

We analyze the differential equations produced by the method of creative telescoping applied to a hyperexponential term in two variables. We show that equations of low order have high degree, and that higher order equations have lower degree. More precisely, we derive degree bounding formulas which allow to estimate the degree of the output equations from creative telescoping as a function of the order. As an application, we show how the knowledge of these formulas can be used to improve, at least in principle, the performance of creative telescoping implementations, and we deduce bounds on the asymptotic complexity of creative telescoping for hyperexponential terms.

We give an overview over computer algebra algorithms for dealing with symbolic sums, recurrence equations, generating functions, and asymptotic estimates, and we will illustrate how to apply these algorithms to problems arising in discrete mathematics.

We continue to investigate which polynomials can possibly occur as factors in the denominators of rational solutions of a given partial linear difference equation. In an earlier article we have introduced the distinction between periodic and aperiodic factors in the denominator, and we have given an algorithm for predicting the aperiodic ones. Now we extend this technique towards the periodic case and present a refined algorithm which also finds most of the periodic factors.

We consider two algorithms which can be used for proving positivity of sequences that are defined by a linear recurrence equation with polynomial coefficients (P-finite sequences). Both algorithms have in common that while they do succeed on a great many examples, there is no guarantee for them to terminate, and they do in fact not terminate for every input. For some restricted classes of P-finite recurrence equations of order up to three we provide a priori criteria that assert the termination of the algorithms.

We investigate which polynomials can possibly occur as factors in the denominators of rational solutions of a given partial linear difference equation (PLDE). Two kinds of polynomials are to be distinguished, we call them periodic and aperiodic. The main result is a generalization of a well-known denominator bounding technique for univariate equations to PLDEs. This generalization is able to find all the aperiodic factors of the denominators for a given PLDE.

We extend Zeilberger's approach to special function identities to cases that are not holonomic. The method of creative telescoping is thus applied to definite sums or integrals involving Stirling or Bernoulli numbers, incomplete Gamma function or polylogarithms, which are not covered by the holonomic framework. The basic idea is to take into account the dimension of appropriate ideals in Ore algebras. This unifies several earlier extensions and provides algorithms for summation and integration in classes that had not been accessible to computer algebra before.

A new method is proposed for finding the logarithmic part of an integral over an algebraic function. The method uses Groebner bases and is easy to implement. It does not have the feature of finding a closed form of an integral whenever there is one. But it very often does, as we will show by a comparison with the built-in integrators of some computer algebra systems.

An extension of Karr’s summation algorithm is presented by which symbolic sums involving radical expressions can be simplified. We discuss the construction of appropriate difference fields as well as algorithms for solving difference equations in these fields. The paper is concluded by a list of identities found with an implementation of our techniques.

We introduce the class of nested polynomially recurrent sequences which includes a large number of sequences that are of combinatorial interest. We present an algorithm for deciding zero equivalence of these sequences, thereby providing a new algorithm for proving identities among combinatorial sequences: In order to prove an identity, decide by the algorithm whether the difference of lefthand-side and righthand-side is identically zero. This algorithm is able to treat mathematical objects which are not covered by any other known symbolic method for proving combinatorial identities. Despite its theoretical flavor and high complexity, an implementation of the algorithm can be successfully applied to nontrivial examples.

We consider symbolic sums which contain subexpressions representing unspecified sequences. Existing symbolic summation technology is extended to sums of this kind. We show how this can be applied in the systematic search for general summation identities. Both, results about the non-existence of identities of a certain form, and examples of general families of identities which we have discovered automatically are included in the paper.

We define a class of special function inequalities that contains many classical examples, such as the Cauchy-Schwarz inequality, and introduce a proving procedure based on induction and Cylindrical Algebraic Decomposition. We present an array of non-trivial examples that can be done by our method. Most of them have not been proven automatically before. Some difficult well-known inequalities such as the Askey-Gasper inequality and Vietoris's inequality lie in our class as well, but we do not know if our proving procedure terminates for them.

Categories and Subject Descriptors I.1.2 [Computing Methodologies]: Symbolic and Algebraic Manipulation--Algorithms