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.
{"title":"Order-Degree-Height Surfaces for Linear Operators","authors":"Hui Huang, Manuel Kauers, G. Mukherjee","doi":"10.1145/3476446.3536187","DOIUrl":"https://doi.org/10.1145/3476446.3536187","url":null,"abstract":"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.","PeriodicalId":130499,"journal":{"name":"Proceedings of the 2022 International Symposium on Symbolic and Algebraic Computation","volume":"11 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-05-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129862333","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let K be a field of characteristic zero and K[x1,...,xn] the corresponding multivariate polynomial ring. Given a sequence of s polynomials f = (f_1,...,f_s) and a polynomial φ, all in K[x1,...,xn] with s>n, we consider the problem of computing the set W(φ,f ) of points at which f vanishes and the Jacobian matrix of f, φ with respect to x1,...,xn does not have full rank. This problem plays an essential role in many application areas.
{"title":"Computing Critical Points for Algebraic Systems Defined by Hyperoctahedral Invariant Polynomials","authors":"Thi Xuan Vu","doi":"10.1145/3476446.3536181","DOIUrl":"https://doi.org/10.1145/3476446.3536181","url":null,"abstract":"Let K be a field of characteristic zero and K[x1,...,xn] the corresponding multivariate polynomial ring. Given a sequence of s polynomials f = (f_1,...,f_s) and a polynomial φ, all in K[x1,...,xn] with s>n, we consider the problem of computing the set W(φ,f ) of points at which f vanishes and the Jacobian matrix of f, φ with respect to x1,...,xn does not have full rank. This problem plays an essential role in many application areas.","PeriodicalId":130499,"journal":{"name":"Proceedings of the 2022 International Symposium on Symbolic and Algebraic Computation","volume":"11 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-03-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132555273","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
D. Chablat, R'emi Pr'ebet, M. S. E. Din, Durgesh Haribhau Salunkhe, P. Wenger
Cuspidal robots are robots with at least two inverse kinematic solutions that can be connected by a singularity-free path. Deciding the cuspidality of generic 3R robots has been studied in the past, but extending the study to six-degree-of-freedom robots can be a challenging problem. Many robots can be modeled as a polynomial map together with a real algebraic set so that the notion of cuspidality can be extended to these data. In this paper we design an algorithm that, on input a polynomial map in n indeterminates, and s polynomials in the same indeterminates describing a real algebraic set of dimension d, decides the cuspidality of the restriction of the map to the real algebraic set under consideration. Moreover, if D and τ are, respectively the maximum degree and the bound on the bit size of the coefficients of the input polynomials, this algorithm runs in time log-linear in τ and polynomial in ((s+d)D)O(n2). It relies on many high-level algorithms in computer algebra which use advanced methods on real algebraic sets and critical loci of polynomial maps. As far as we know, this is the first algorithm that tackles the cuspidality problem from a general point of view.
{"title":"Deciding Cuspidality of Manipulators through Computer Algebra and Algorithms in Real Algebraic Geometry","authors":"D. Chablat, R'emi Pr'ebet, M. S. E. Din, Durgesh Haribhau Salunkhe, P. Wenger","doi":"10.1145/3476446.3535477","DOIUrl":"https://doi.org/10.1145/3476446.3535477","url":null,"abstract":"Cuspidal robots are robots with at least two inverse kinematic solutions that can be connected by a singularity-free path. Deciding the cuspidality of generic 3R robots has been studied in the past, but extending the study to six-degree-of-freedom robots can be a challenging problem. Many robots can be modeled as a polynomial map together with a real algebraic set so that the notion of cuspidality can be extended to these data. In this paper we design an algorithm that, on input a polynomial map in n indeterminates, and s polynomials in the same indeterminates describing a real algebraic set of dimension d, decides the cuspidality of the restriction of the map to the real algebraic set under consideration. Moreover, if D and τ are, respectively the maximum degree and the bound on the bit size of the coefficients of the input polynomials, this algorithm runs in time log-linear in τ and polynomial in ((s+d)D)O(n2). It relies on many high-level algorithms in computer algebra which use advanced methods on real algebraic sets and critical loci of polynomial maps. As far as we know, this is the first algorithm that tackles the cuspidality problem from a general point of view.","PeriodicalId":130499,"journal":{"name":"Proceedings of the 2022 International Symposium on Symbolic and Algebraic Computation","volume":"178 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-03-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121899719","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
P. Kutas, Mickaël Montessinos, Gergely Zábrádi, T'imea Csah'ok
We propose polynomial-time algorithms for finding nontrivial zeros of quadratic forms with four variables over rational function fields of characteristic 2. We apply these results to find prescribed quadratic subfields of quaternion division algebras and zero divisors in $M_2(D)$, the full matrix algebra over a division algebra, given by structure constants. We also provide an implementation of our results in MAGMA which shows that the algorithms are truly practical.
{"title":"Finding Nontrivial Zeros of Quadratic Forms over Rational Function Fields of Characteristic 2","authors":"P. Kutas, Mickaël Montessinos, Gergely Zábrádi, T'imea Csah'ok","doi":"10.1145/3476446.3535485","DOIUrl":"https://doi.org/10.1145/3476446.3535485","url":null,"abstract":"We propose polynomial-time algorithms for finding nontrivial zeros of quadratic forms with four variables over rational function fields of characteristic 2. We apply these results to find prescribed quadratic subfields of quaternion division algebras and zero divisors in $M_2(D)$, the full matrix algebra over a division algebra, given by structure constants. We also provide an implementation of our results in MAGMA which shows that the algorithms are truly practical.","PeriodicalId":130499,"journal":{"name":"Proceedings of the 2022 International Symposium on Symbolic and Algebraic Computation","volume":"43 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-03-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134278046","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Real-world phenomena can often be conveniently described by dynamical systems (that is, ODE systems in the state-space form). However, if one observes the state of the system only partially, the observed quantities (outputs) and the inputs of the system can typically be related by more complicated differential-algebraic equations (DAEs). Therefore, a natural question (referred to as the realizability problem) is: given a differential-algebraic equation (say, fitted from data), does it come from a partially observed dynamical system? A special case in which the functions involved in the dynamical system are rational is of particular interest. For a single differential-algebraic equation in a single output variable, Forsman has shown that it is realizable by a rational dynamical system if and only if the corresponding hypersurface is unirational, and he turned this into an algorithm in the first-order case. In this paper, we study a more general case of single-input-single-output equations. We show that if a realization by a rational dynamical system exists, the system can be taken to have the dimension equal to the order of the DAE. We provide a complete algorithm for first-order DAEs. We also show that the same approach can be used for higher-order DAEs using several examples from the literature.
{"title":"On Realizing Differential-Algebraic Equations by Rational Dynamical Systems","authors":"D. Pavlov, G. Pogudin","doi":"10.1145/3476446.3535492","DOIUrl":"https://doi.org/10.1145/3476446.3535492","url":null,"abstract":"Real-world phenomena can often be conveniently described by dynamical systems (that is, ODE systems in the state-space form). However, if one observes the state of the system only partially, the observed quantities (outputs) and the inputs of the system can typically be related by more complicated differential-algebraic equations (DAEs). Therefore, a natural question (referred to as the realizability problem) is: given a differential-algebraic equation (say, fitted from data), does it come from a partially observed dynamical system? A special case in which the functions involved in the dynamical system are rational is of particular interest. For a single differential-algebraic equation in a single output variable, Forsman has shown that it is realizable by a rational dynamical system if and only if the corresponding hypersurface is unirational, and he turned this into an algorithm in the first-order case. In this paper, we study a more general case of single-input-single-output equations. We show that if a realization by a rational dynamical system exists, the system can be taken to have the dimension equal to the order of the DAE. We provide a complete algorithm for first-order DAEs. We also show that the same approach can be used for higher-order DAEs using several examples from the literature.","PeriodicalId":130499,"journal":{"name":"Proceedings of the 2022 International Symposium on Symbolic and Algebraic Computation","volume":"11 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-03-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"117002972","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Apostolos Chalkis, Christina Katsamaki, Josué Tonelli-Cueto
Given a parametric polynomial curve γ:[a,b] →Rn, how can we sample a random point x ∈ im(γ) in such a way that it is distributed uniformly with respect to the arc-length? Unfortunately, we cannot sample exactly such a point---even assuming we can perform exact arithmetic operations. So we end up with the following question: how does the method we choose affect the quality of the approximate sample we obtain? In practice, there are many answers. However, in theory, there are still gaps in our understanding. In this paper, we address this question from the point of view of complexity theory, providing bounds in terms of the size of the desired error.
{"title":"On the Error of Random Sampling: Uniformly Distributed Random Points on Parametric Curves","authors":"Apostolos Chalkis, Christina Katsamaki, Josué Tonelli-Cueto","doi":"10.1145/3476446.3536190","DOIUrl":"https://doi.org/10.1145/3476446.3536190","url":null,"abstract":"Given a parametric polynomial curve γ:[a,b] →Rn, how can we sample a random point x ∈ im(γ) in such a way that it is distributed uniformly with respect to the arc-length? Unfortunately, we cannot sample exactly such a point---even assuming we can perform exact arithmetic operations. So we end up with the following question: how does the method we choose affect the quality of the approximate sample we obtain? In practice, there are many answers. However, in theory, there are still gaps in our understanding. In this paper, we address this question from the point of view of complexity theory, providing bounds in terms of the size of the desired error.","PeriodicalId":130499,"journal":{"name":"Proceedings of the 2022 International Symposium on Symbolic and Algebraic Computation","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-03-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130757954","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pascal Giorgi, Bruno Grenet, Armelle Perret du Cray, Daniel S. Roche
Numerous algorithms call for computation over the integers modulo a randomly-chosen large prime. In some cases, the quasi-cubic complexity of selecting a random prime can dominate the total running time. We propose a new variant of dynamic evaluation, applied to a randomly-chosen (composite) integer. The transformation we propose can apply to any algorithm in the algebraic RAM model, even allowing randomization. The resulting transformed algorithm avoids any primality tests and will, with constant positive probability, have the same result as the original computation modulo a randomly-chosen prime. As an application, we demonstrate how to compute the exact number of nonzero terms in an unknown integer polynomial in quasi-linear time. We also show how the same algorithmic transformation technique can be used for computing modulo random irreducible polynomials over a finite field.
{"title":"Random Primes without Primality Testing","authors":"Pascal Giorgi, Bruno Grenet, Armelle Perret du Cray, Daniel S. Roche","doi":"10.1145/3476446.3536191","DOIUrl":"https://doi.org/10.1145/3476446.3536191","url":null,"abstract":"Numerous algorithms call for computation over the integers modulo a randomly-chosen large prime. In some cases, the quasi-cubic complexity of selecting a random prime can dominate the total running time. We propose a new variant of dynamic evaluation, applied to a randomly-chosen (composite) integer. The transformation we propose can apply to any algorithm in the algebraic RAM model, even allowing randomization. The resulting transformed algorithm avoids any primality tests and will, with constant positive probability, have the same result as the original computation modulo a randomly-chosen prime. As an application, we demonstrate how to compute the exact number of nonzero terms in an unknown integer polynomial in quasi-linear time. We also show how the same algorithmic transformation technique can be used for computing modulo random irreducible polynomials over a finite field.","PeriodicalId":130499,"journal":{"name":"Proceedings of the 2022 International Symposium on Symbolic and Algebraic Computation","volume":"69 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-02-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128770236","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Aircraft models may be considered as flat if one neglects some terms associated to aerodynamics. Computational experiments in Maple show that in some cases a suitably designed feed-back allows to follow such trajectories, when applied to the non-flat model. However some maneuvers may be hard or even impossible to achieve with this flat approximation. In this paper, we propose an iterated process to compute a more achievable trajectory, starting from the flat reference trajectory. More precisely, the unknown neglected terms in the flat model are iteratively re-evaluated using the values obtained at the previous step. This process may be interpreted as a new trajectory parametrization, using an infinite number of derivatives, a property that may be called generalized flatness. We illustrate the pertinence of this approach in flight conditions of increasing difficulties, from single engine flight, to aileron roll.
{"title":"Extending Flat Motion Planning to Non-flat Systems. Experiments on Aircraft Models Using Maple","authors":"F. Ollivier","doi":"10.1145/3476446.3536179","DOIUrl":"https://doi.org/10.1145/3476446.3536179","url":null,"abstract":"Aircraft models may be considered as flat if one neglects some terms associated to aerodynamics. Computational experiments in Maple show that in some cases a suitably designed feed-back allows to follow such trajectories, when applied to the non-flat model. However some maneuvers may be hard or even impossible to achieve with this flat approximation. In this paper, we propose an iterated process to compute a more achievable trajectory, starting from the flat reference trajectory. More precisely, the unknown neglected terms in the flat model are iteratively re-evaluated using the values obtained at the previous step. This process may be interpreted as a new trajectory parametrization, using an infinite number of derivatives, a property that may be called generalized flatness. We illustrate the pertinence of this approach in flight conditions of increasing difficulties, from single engine flight, to aileron roll.","PeriodicalId":130499,"journal":{"name":"Proceedings of the 2022 International Symposium on Symbolic and Algebraic Computation","volume":"13 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-02-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124774626","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We construct Mahler discrete residues for rational functions and show that they comprise a complete obstruction to the Mahler summability problem of deciding whether a given rational function $f(x)$ is of the form $g(x^p)-g(x)$ for some rational function $g(x)$ and an integer $p > 1$. This extends to the Mahler case the analogous notions, properties, and applications of discrete residues (in the shift case) and q-discrete residues (in the q-difference case) developed by Chen and Singer. Along the way we define several additional notions that promise to be useful for addressing related questions involving Mahler difference fields of rational functions, including in particular telescoping problems and problems in the (differential) Galois theory of Mahler difference equations.
{"title":"Mahler Discrete Residues and Summability for Rational Functions","authors":"Carlos E. Arreche, Yi Zhang","doi":"10.1145/3476446.3536186","DOIUrl":"https://doi.org/10.1145/3476446.3536186","url":null,"abstract":"We construct Mahler discrete residues for rational functions and show that they comprise a complete obstruction to the Mahler summability problem of deciding whether a given rational function $f(x)$ is of the form $g(x^p)-g(x)$ for some rational function $g(x)$ and an integer $p > 1$. This extends to the Mahler case the analogous notions, properties, and applications of discrete residues (in the shift case) and q-discrete residues (in the q-difference case) developed by Chen and Singer. Along the way we define several additional notions that promise to be useful for addressing related questions involving Mahler difference fields of rational functions, including in particular telescoping problems and problems in the (differential) Galois theory of Mahler difference equations.","PeriodicalId":130499,"journal":{"name":"Proceedings of the 2022 International Symposium on Symbolic and Algebraic Computation","volume":"189 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-02-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121096871","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Solving zero-dimensional polynomial systems using Gröbner bases is usually done by, first, computing a Gröbner basis for the degree reverse lexicographic order, and next computing the lexicographic Gröbner basis with a change of order algorithm. Currently, the change of order now takes a significant part of the whole solving time for many generic instances. Like the fastest known change of order algorithms, this work focuses on the situation where the ideal defined by the system satisfies natural properties which can be recovered in generic coordinates. First, the ideal has a shape lexicographic Gröbner basis. Second, the set of leading terms with respect to the degree reverse lexicographic order has a stability property; in particular, the multiplication matrix can be read on the input Gröbner basis. The current fastest algorithms rely on the sparsity of this matrix. Actually, this sparsity is a consequence of an algebraic structure, which can be exploited to represent the matrix concisely as a univariate polynomial matrix. We show that the Hermite normal form of that matrix yields the sought lexicographic Gröbner basis, under assumptions which cover the shape position case. Under some mild assumption implying n≤t, the arithmetic complexity of our algorithm is O~(tω-1D), where n is the number of variables, t is a sparsity indicator of the aforementioned matrix, D is the degree of the zero-dimensional ideal under consideration, and ω is the exponent of matrix multiplication. This improves upon both state-of-the-art complexity bounds O~(tD2) and O~(Dω, since ω<3 and t≤D. Practical experiments, based on the libraries msolve and PML, confirm the high practical benefit.
{"title":"Faster Change of Order Algorithm for Gröbner Bases under Shape and Stability Assumptions","authors":"Jérémy Berthomieu, Vincent Neiger, M. S. E. Din","doi":"10.1145/3476446.3535484","DOIUrl":"https://doi.org/10.1145/3476446.3535484","url":null,"abstract":"Solving zero-dimensional polynomial systems using Gröbner bases is usually done by, first, computing a Gröbner basis for the degree reverse lexicographic order, and next computing the lexicographic Gröbner basis with a change of order algorithm. Currently, the change of order now takes a significant part of the whole solving time for many generic instances. Like the fastest known change of order algorithms, this work focuses on the situation where the ideal defined by the system satisfies natural properties which can be recovered in generic coordinates. First, the ideal has a shape lexicographic Gröbner basis. Second, the set of leading terms with respect to the degree reverse lexicographic order has a stability property; in particular, the multiplication matrix can be read on the input Gröbner basis. The current fastest algorithms rely on the sparsity of this matrix. Actually, this sparsity is a consequence of an algebraic structure, which can be exploited to represent the matrix concisely as a univariate polynomial matrix. We show that the Hermite normal form of that matrix yields the sought lexicographic Gröbner basis, under assumptions which cover the shape position case. Under some mild assumption implying n≤t, the arithmetic complexity of our algorithm is O~(tω-1D), where n is the number of variables, t is a sparsity indicator of the aforementioned matrix, D is the degree of the zero-dimensional ideal under consideration, and ω is the exponent of matrix multiplication. This improves upon both state-of-the-art complexity bounds O~(tD2) and O~(Dω, since ω<3 and t≤D. Practical experiments, based on the libraries msolve and PML, confirm the high practical benefit.","PeriodicalId":130499,"journal":{"name":"Proceedings of the 2022 International Symposium on Symbolic and Algebraic Computation","volume":"75 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-02-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132959416","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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