We address the problem of computing a drawing of high resolution of a plane curve defined by a bivariate polynomial equation P(x,y)=0. Given a grid of fixed resolution, a drawing is a subset of pixels. Our goal is to compute an approximate drawing that (i) contains all the parts of the curve that intersect the pixel edges, (ii) excludes a pixel when the evaluation of P with interval arithmetic on each of its four edges is far from zero. One of the challenges for computing drawings on a high-resolution grid is to minimize the complexity due to the evaluation of the input polynomial. Most state-of-the-art approaches focus on bounding the number of independent evaluations. Using state-of-the-art Computer Algebra techniques, we design new algorithms that amortize the evaluations and improve the complexity for computing such drawings. Our main contribution is to use a non-uniform grid based on the Chebyshev nodes to take advantage of multipoint evaluation techniques via the Discrete Cosine Transform. We propose two new algorithms that compute drawings and compare them experimentally on several classes of high degree polynomials. Notably, one of those approaches is faster than state-of-the-art drawing software.
{"title":"Fast High-Resolution Drawing of Algebraic Curves","authors":"Nuwan Herath Mudiyanselage, G. Moroz, M. Pouget","doi":"10.1145/3476446.3535483","DOIUrl":"https://doi.org/10.1145/3476446.3535483","url":null,"abstract":"We address the problem of computing a drawing of high resolution of a plane curve defined by a bivariate polynomial equation P(x,y)=0. Given a grid of fixed resolution, a drawing is a subset of pixels. Our goal is to compute an approximate drawing that (i) contains all the parts of the curve that intersect the pixel edges, (ii) excludes a pixel when the evaluation of P with interval arithmetic on each of its four edges is far from zero. One of the challenges for computing drawings on a high-resolution grid is to minimize the complexity due to the evaluation of the input polynomial. Most state-of-the-art approaches focus on bounding the number of independent evaluations. Using state-of-the-art Computer Algebra techniques, we design new algorithms that amortize the evaluations and improve the complexity for computing such drawings. Our main contribution is to use a non-uniform grid based on the Chebyshev nodes to take advantage of multipoint evaluation techniques via the Discrete Cosine Transform. We propose two new algorithms that compute drawings and compare them experimentally on several classes of high degree polynomials. Notably, one of those approaches is faster than state-of-the-art drawing software.","PeriodicalId":130499,"journal":{"name":"Proceedings of the 2022 International Symposium on Symbolic and Algebraic Computation","volume":"11 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116899372","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 give methods to develop efficiently computable bijections between the rational numbers and the positive integers. That is, given a rational number in the standard representation a/b, where a, b are integers, we can compute n, its position in the enumeration, in time polynomial in the bit lengths of a and b. Conversely, given a position n in the enumeration, we can compute the corresponding rational number a/b at that position in time polynomial in the bit length of n. This is not the first such bijection to have appeared in the literature. However, we submit that the method presented here, which uses König's proof of the Schröder-Bernstein Theorem, is relatively simple to understand, and has a broad application. It can be applied to enumerating other denumerable sets. As an example, we use it to give a polynomial-time bijection between the algebraic numbers and the positive integers.
{"title":"Enumerating Denumerable Sets in Polynomial Time via the Schröder--Bernstein Theorem","authors":"Reinhold Burger","doi":"10.1145/3476446.3535479","DOIUrl":"https://doi.org/10.1145/3476446.3535479","url":null,"abstract":"We give methods to develop efficiently computable bijections between the rational numbers and the positive integers. That is, given a rational number in the standard representation a/b, where a, b are integers, we can compute n, its position in the enumeration, in time polynomial in the bit lengths of a and b. Conversely, given a position n in the enumeration, we can compute the corresponding rational number a/b at that position in time polynomial in the bit length of n. This is not the first such bijection to have appeared in the literature. However, we submit that the method presented here, which uses König's proof of the Schröder-Bernstein Theorem, is relatively simple to understand, and has a broad application. It can be applied to enumerating other denumerable sets. As an example, we use it to give a polynomial-time bijection between the algebraic numbers and the positive integers.","PeriodicalId":130499,"journal":{"name":"Proceedings of the 2022 International Symposium on Symbolic and Algebraic Computation","volume":"112 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124807938","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}
Given an autonomous first order algebraic ordinary differential equation $F(y,y')=0$, we provide algorithms for computing formal Puiseux series solutions of $F(y,y')=0$ with real or rational coefficients. For this purpose we give necessary and sufficient conditions on the existence of such solutions by combining classical methods from algebraic geometry and the study of an associated differential equation. Since all formal Puiseux series solutions of such differential equations are convergent in a certain neighborhood, the solutions also define real solution functions.
{"title":"Puiseux Series Solutions with Real or Rational Coefficients of First Order Autonomous AODEs","authors":"Sebastian Falkensteiner","doi":"10.1145/3476446.3536185","DOIUrl":"https://doi.org/10.1145/3476446.3536185","url":null,"abstract":"Given an autonomous first order algebraic ordinary differential equation $F(y,y')=0$, we provide algorithms for computing formal Puiseux series solutions of $F(y,y')=0$ with real or rational coefficients. For this purpose we give necessary and sufficient conditions on the existence of such solutions by combining classical methods from algebraic geometry and the study of an associated differential equation. Since all formal Puiseux series solutions of such differential equations are convergent in a certain neighborhood, the solutions also define real solution functions.","PeriodicalId":130499,"journal":{"name":"Proceedings of the 2022 International Symposium on Symbolic and Algebraic Computation","volume":"115 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116103563","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}
My lecture will survey some classical and recent validated computing algorithms based on the theory of set-valued analysis, in suitable functional spaces, as well as by combining symbolic and numerical computations. These techniques are illustrated with some applications which appear in practical space mission analysis and design. This is only a short summary of the talk.
{"title":"Validated Numerics: Algorithms and Practical Applications in Aerospace","authors":"Mioara Joldes","doi":"10.1145/3476446.3535505","DOIUrl":"https://doi.org/10.1145/3476446.3535505","url":null,"abstract":"My lecture will survey some classical and recent validated computing algorithms based on the theory of set-valued analysis, in suitable functional spaces, as well as by combining symbolic and numerical computations. These techniques are illustrated with some applications which appear in practical space mission analysis and design. This is only a short summary of the talk.","PeriodicalId":130499,"journal":{"name":"Proceedings of the 2022 International Symposium on Symbolic and Algebraic Computation","volume":"34 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128108325","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 present Hermite polynomial interpolation algorithms that for a sparse univariate polynomial f with coefficients from a field compute the polynomial from fewer points than the classical algorithms. If the interpolating polynomial f has t terms, our algorithms, which use randomization, require argument/value triples (wi,f(wi),f'(wi)) for i=0, ..., t + ↾(t+1)/2↿ - 1, where w is randomly sampled and the probability of a correct output is determined from a degree bound for f. With f' we denote the derivative of f. Our algorithms generalize to multivariate polynomials, higher derivatives and sparsity with respect to Chebyshev polynomial bases. We have algorithms that can correct errors in the points by oversampling at a limited number of good values. If an upper bound B ≥ t for the number of terms is given, our algorithms use a randomly selected w and, with high probability, t/2 + B triples, but then never return an incorrect output.
{"title":"Sparse Polynomial Hermite Interpolation","authors":"E. Kaltofen","doi":"10.1145/3476446.3535501","DOIUrl":"https://doi.org/10.1145/3476446.3535501","url":null,"abstract":"We present Hermite polynomial interpolation algorithms that for a sparse univariate polynomial f with coefficients from a field compute the polynomial from fewer points than the classical algorithms. If the interpolating polynomial f has t terms, our algorithms, which use randomization, require argument/value triples (wi,f(wi),f'(wi)) for i=0, ..., t + ↾(t+1)/2↿ - 1, where w is randomly sampled and the probability of a correct output is determined from a degree bound for f. With f' we denote the derivative of f. Our algorithms generalize to multivariate polynomials, higher derivatives and sparsity with respect to Chebyshev polynomial bases. We have algorithms that can correct errors in the points by oversampling at a limited number of good values. If an upper bound B ≥ t for the number of terms is given, our algorithms use a randomly selected w and, with high probability, t/2 + B triples, but then never return an incorrect output.","PeriodicalId":130499,"journal":{"name":"Proceedings of the 2022 International Symposium on Symbolic and Algebraic Computation","volume":"151 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124058436","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}
Rodrigo Iglesias, Patricia Pascual-Ortigosa, E. Sáenz-de-Cabezón
The evaluation of system reliability is an NP-hard problem even in the binary case. There exist several general methodologies to analyze and compute system reliability. The two main ones are the sum-of-disjoint-products (SDP), which expresses the logic function of the system as a union of disjoint terms, and the Improved Inclusion-Exclusion (IIE) formulas. The algebraic approach to system reliability, assigns a monomial ideal to the system and computes its reliability in terms of the Hilbert series of the ideal, providing an algebraic version of the IIE method. In this paper we make use of this monomial ideal framework and present an algebraic version of the SDP method, based on a combinatorial decomposition of the system's ideal. Such a decomposition is obtained from an involutive basis of the ideal. This algebraic version is suitable for binary and multi-state systems. We include computer experiments on the performance of this approach using the C++ computer algebra library CoCoALib and a discussion on which of the algebraic methods can be more efficient depending on the type of system under analysis.
{"title":"An Algebraic Version of the Sum-of-disjoint-products Method for Multi-state System Reliability Analysis","authors":"Rodrigo Iglesias, Patricia Pascual-Ortigosa, E. Sáenz-de-Cabezón","doi":"10.1145/3476446.3535472","DOIUrl":"https://doi.org/10.1145/3476446.3535472","url":null,"abstract":"The evaluation of system reliability is an NP-hard problem even in the binary case. There exist several general methodologies to analyze and compute system reliability. The two main ones are the sum-of-disjoint-products (SDP), which expresses the logic function of the system as a union of disjoint terms, and the Improved Inclusion-Exclusion (IIE) formulas. The algebraic approach to system reliability, assigns a monomial ideal to the system and computes its reliability in terms of the Hilbert series of the ideal, providing an algebraic version of the IIE method. In this paper we make use of this monomial ideal framework and present an algebraic version of the SDP method, based on a combinatorial decomposition of the system's ideal. Such a decomposition is obtained from an involutive basis of the ideal. This algebraic version is suitable for binary and multi-state systems. We include computer experiments on the performance of this approach using the C++ computer algebra library CoCoALib and a discussion on which of the algebraic methods can be more efficient depending on the type of system under analysis.","PeriodicalId":130499,"journal":{"name":"Proceedings of the 2022 International Symposium on Symbolic and Algebraic Computation","volume":"66 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128453592","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}
The Canny-Emiris formula [3] gives the sparse resultant as a ratio between the determinant of a Sylvester-type matrix and a minor of it, by a subdivision algorithm. The most complete proof of the formula was given by D'Andrea et al. in [9] under general conditions on the underlying mixed subdivision. Before the proof, Canny and Pedersen had proposed [5] a greedy algorithm which provides smaller matrices, in general. The goal of this paper is to give an explicit class of mixed subdivisions for the greedy approach such that the formula holds, and the dimensions of the matrices are reduced compared to the subdivision algorithm. We measure this reduction for the case when the Newton polytopes are zonotopes generated by n line segments (where n is the rank of the underlying lattice), and for the case of multihomogeneous systems. This article comes with a JULIA implementation of the treated cases.
{"title":"A Greedy Approach to the Canny-Emiris Formula","authors":"Carles Checa, I. Emiris","doi":"10.1145/3476446.3536180","DOIUrl":"https://doi.org/10.1145/3476446.3536180","url":null,"abstract":"The Canny-Emiris formula [3] gives the sparse resultant as a ratio between the determinant of a Sylvester-type matrix and a minor of it, by a subdivision algorithm. The most complete proof of the formula was given by D'Andrea et al. in [9] under general conditions on the underlying mixed subdivision. Before the proof, Canny and Pedersen had proposed [5] a greedy algorithm which provides smaller matrices, in general. The goal of this paper is to give an explicit class of mixed subdivisions for the greedy approach such that the formula holds, and the dimensions of the matrices are reduced compared to the subdivision algorithm. We measure this reduction for the case when the Newton polytopes are zonotopes generated by n line segments (where n is the rank of the underlying lattice), and for the case of multihomogeneous systems. This article comes with a JULIA implementation of the treated cases.","PeriodicalId":130499,"journal":{"name":"Proceedings of the 2022 International Symposium on Symbolic and Algebraic Computation","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129113751","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}
Quadratic modules in real algebraic geometry are akin to polynomial ideals in algebraic geometry, and have been found useful in the theory of Positivstellensatz to study Hilbert's 17th problem. Algorithms are presented in this paper for testing membership in univariate finitely generated quadratic modules over the reals and inclusion of two finitely generated quadratic modules. For a univariate unbounded quadratic module, an explicit upper bound on the degrees of sums of squares to construct any given polynomial is proved and then used to design an algorithm for testing membership in such a quadratic module. For a bounded quadratic module, a unique signature is associated with it based on the real values on which its finite basis is non-negative, and the signatures are used to furnish a criterion for inclusion of two finitely generated quadratic modules and a corresponding algorithm which solves the membership problem as a special case. It is also shown that a bounded quadratic module can be transformed to an equivalent one with two generators with an algorithm for performing this transformation. All the presented algorithms have been implemented.
{"title":"Algorithms for Testing Membership in Univariate Quadratic Modules over the Reals","authors":"W. Shang, Chenqi Mou, D. Kapur","doi":"10.1145/3476446.3536176","DOIUrl":"https://doi.org/10.1145/3476446.3536176","url":null,"abstract":"Quadratic modules in real algebraic geometry are akin to polynomial ideals in algebraic geometry, and have been found useful in the theory of Positivstellensatz to study Hilbert's 17th problem. Algorithms are presented in this paper for testing membership in univariate finitely generated quadratic modules over the reals and inclusion of two finitely generated quadratic modules. For a univariate unbounded quadratic module, an explicit upper bound on the degrees of sums of squares to construct any given polynomial is proved and then used to design an algorithm for testing membership in such a quadratic module. For a bounded quadratic module, a unique signature is associated with it based on the real values on which its finite basis is non-negative, and the signatures are used to furnish a criterion for inclusion of two finitely generated quadratic modules and a corresponding algorithm which solves the membership problem as a special case. It is also shown that a bounded quadratic module can be transformed to an equivalent one with two generators with an algorithm for performing this transformation. All the presented algorithms have been implemented.","PeriodicalId":130499,"journal":{"name":"Proceedings of the 2022 International Symposium on Symbolic and Algebraic Computation","volume":"117 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121478471","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 improve significantly the Nart-Montes algorithm for factoring polynomials over a complete discrete valuation ring A. Our first contribution is to extend the Hensel lemma in the context of generalised Newton polygons, from which we derive a new divide and conquer strategy. Also, if A has residual characteristic zero or high enough, we prove that approximate roots are convenient representatives of types, leading finally to an almost optimal complexity both for irreducibility and factorisation issues, plus the cost of factorisations above the residue field. For instance, to compute an OM-factorisation of F∈A[x], we improve the complexity results of [3] by a factor δ, the discriminant valuation of F.
{"title":"Local Polynomial Factorisation: Improving the Montes Algorithm","authors":"A. Poteaux, Martin Weimann","doi":"10.1145/3476446.3535487","DOIUrl":"https://doi.org/10.1145/3476446.3535487","url":null,"abstract":"We improve significantly the Nart-Montes algorithm for factoring polynomials over a complete discrete valuation ring A. Our first contribution is to extend the Hensel lemma in the context of generalised Newton polygons, from which we derive a new divide and conquer strategy. Also, if A has residual characteristic zero or high enough, we prove that approximate roots are convenient representatives of types, leading finally to an almost optimal complexity both for irreducibility and factorisation issues, plus the cost of factorisations above the residue field. For instance, to compute an OM-factorisation of F∈A[x], we improve the complexity results of [3] by a factor δ, the discriminant valuation of F.","PeriodicalId":130499,"journal":{"name":"Proceedings of the 2022 International Symposium on Symbolic and Algebraic Computation","volume":"93 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122756684","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 present a new algorithm for performing linear Hensel lifting on bivariate polynomials over the finite field Zp for some prime p. Our algorithm lifts n monic, univariate polynomials to recover the factors of a polynomial A(x,y) in Zp[x,y] which is monic in x, and bounded by degrees dx = deg(A,x) and dy = deg(A,y). Our algorithm improves upon Bernardin's algorithm in [1] and reduces the number of arithmetic operations in Zp from O(n dx^2 dy^2) to O(dx^2 dy + dx dy^2) for p >= dx. Experimental results in C verify that our algorithm compares favorably with Bernardin's for large degree polynomials. Moreover, we've implemented a Quadratic Hensel lifting algorithm in Magma to show that our cubic Linear Hensel lifting algorithm outperforms Magma's Quadratic Hensel lifting for a wide range of input sizes.
我们提出了一种新的算法，用于在有限域Zp上对某些素数p的二元多项式进行线性Hensel提升。我们的算法提升了n个单变量多项式，以恢复Zp[x,y]中多项式a (x,y)的因子，该多项式在x中是单变量的，并且以度dx = deg(a,x)和dy = deg(a,y)为界。我们的算法改进了[1]中的Bernardin算法，并将Zp中的算术运算次数从O(n dx^2 dy^2)减少到O(dx^2 dy + dx dy^2)， p >= dx。C语言的实验结果验证了我们的算法在处理大次多项式时优于Bernardin算法。此外，我们在Magma中实现了一个二次Hensel提升算法，表明我们的三次线性Hensel提升算法在大范围的输入大小下优于Magma的二次Hensel提升算法。
{"title":"Linear Hensel Lifting for Zp[x,y] for n Factors with Cubic Cost","authors":"M. Monagan, Garrett Paluck","doi":"10.1145/3476446.3536178","DOIUrl":"https://doi.org/10.1145/3476446.3536178","url":null,"abstract":"We present a new algorithm for performing linear Hensel lifting on bivariate polynomials over the finite field Zp for some prime p. Our algorithm lifts n monic, univariate polynomials to recover the factors of a polynomial A(x,y) in Zp[x,y] which is monic in x, and bounded by degrees dx = deg(A,x) and dy = deg(A,y). Our algorithm improves upon Bernardin's algorithm in [1] and reduces the number of arithmetic operations in Zp from O(n dx^2 dy^2) to O(dx^2 dy + dx dy^2) for p >= dx. Experimental results in C verify that our algorithm compares favorably with Bernardin's for large degree polynomials. Moreover, we've implemented a Quadratic Hensel lifting algorithm in Magma to show that our cubic Linear Hensel lifting algorithm outperforms Magma's Quadratic Hensel lifting for a wide range of input sizes.","PeriodicalId":130499,"journal":{"name":"Proceedings of the 2022 International Symposium on Symbolic and Algebraic Computation","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129932697","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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