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Sparse multivariate polynomial interpolation on the basis of Schubert polynomials

Priyanka Mukhopadhyay, Youming Qiao
2016 Computational Complexity  
This generalizes, and derandomizes, the sparse interpolation algorithm of symmetric polynomials in the Schur basis by Barvinok and Fomin (Advances in Applied Mathematics, 18(3):271--285).  ...  These polynomials generalize Schur polynomials, and form a linear basis of multivariate polynomials.  ...  Part of the work was done when Youming was visiting the Simons Institute for the program Algorithms and Complexity in Algebraic Geometry.  ... 
doi:10.1007/s00037-016-0142-y fatcat:vtsm7jboqnbudn47jmq3av75cu

Real solutions to equations from geometry [article]

Frank Sottile
2010 arXiv   pre-print
Understanding, finding, or even deciding on the existence of real solutions to a system of equations is a very difficult problem with many applications.  ...  Such equations from geometry for which we have information about their real solutions are the subject of these lecture notes, which focuses on bounds, both upper and lower, as well as situations in which  ...  One class of systems that we will study are systems of sparse polynomials.  ... 
arXiv:math/0609829v2 fatcat:xqnyznyvsrfbvbe35k2w37rutq

Galois Groups in Enumerative Geometry and Applications [article]

Frank Sottile, Thomas Yahl
2021 arXiv   pre-print
We discuss the current directions of this study, including open problems and conjectures.  ...  Despite this pedigree, the study of Galois groups in enumerative geometry was dormant for a century, with a systematic study only occuring in the past 15 years.  ...  Solving decomposable sparse polynomial systems.  ... 
arXiv:2108.07905v1 fatcat:f4hheo3r3zgynaout3xjo3bsky

An evaluation approach to computing invariants rings of permutation groups [article]

Nicolas Borie, Nicolas M. Thiéry
2011 arXiv   pre-print
Using evaluation at appropriately chosen points, we propose a Gr\"obner basis free approach for calculating the secondary invariants of a finite permutation group.  ...  This approach allows for exploiting the symmetries to confine the calculations into a smaller quotient space, which gives a tighter control on the algorithmic complexity, especially for large groups.  ...  Schubert 362 polynomials or the orderly generation of canonical monomials) [SCc08].  ... 
arXiv:1110.3849v1 fatcat:uuiykugbvrbtrlb7sdqjyb74sy

The Complexity of Computing the Hilbert Polynomial of Smooth Equidimensional Complex Projective Varieties

Peter Burgisser, Martin Lotz
2006 Foundations of Computational Mathematics  
zeros of a finite set of multivariate polynomials.  ...  We continue the study of counting complexity begun in [11, 14, 13] by proving upper and lower bounds on the complexity of computing the Hilbert polynomial of a homogeneous ideal.  ...  We thank Felipe Cucker for discussions and inviting us to Hong Kong in Spring 2004, where the basis of this work was elaborated in the joint papers [14, 13] .  ... 
doi:10.1007/s10208-005-0175-0 fatcat:xndsei6perhvnhalllkpobp4si

Schur Polynomials Do Not Have Small Formulas If the Determinant Doesn't

Prasad Chaugule, Mrinal Kumar, Nutan Limaye, Chandra Kanta Mohapatra, Adrian She, Srikanth Srinivasan, Shubhangi Saraf
2020 Computational Complexity Conference  
To the best of our knowledge, these are one of the first hardness results of this kind for families of polynomials which are not multilinear.  ...  However, unlike some other families of symmetric polynomials like the Elementary Symmetric polynomials, the Power Symmetric polynomials and the Complete Homogeneous Symmetric polynomials, the computational  ...  Similar to the Vandermonde matrix, the determinant of this matrix is related to the problem of sparse polynomial interpolation, where we are trying to interpolate a polynomial only involving the monomials  ... 
doi:10.4230/lipics.ccc.2020.14 dblp:conf/coco/ChauguleKLMS020 fatcat:tvp5gihvbnditghuv4hcf32tm4

Valuative invariants for large classes of matroids [article]

Luis Ferroni, Benjamin Schröter
2022 arXiv   pre-print
This shows that evaluations depend solely on the behavior of the invariant on a well-behaved small subclass of Schubert matroids that we call "cuspidal matroids".  ...  They include the volume and Ehrhart polynomial of base polytopes, the Tutte polynomial, Kazhdan-Lusztig polynomials, the Whitney numbers of the first and second kind, spectrum polynomials and a generalization  ...  This means that for the computation of the coefficients of ehr(Λ , ,ℎ, , ) one has to use Lagrange's interpolation or another method.  ... 
arXiv:2208.04893v2 fatcat:jerpeevawnh7dkdhjzsekqjb5u

The complexity of computing the Hilbert polynomial of smooth equidimensional complex projective varieties [article]

Peter Buergisser, Martin Lotz
2005 arXiv   pre-print
zeros of a finite set of multivariate polynomials.  ...  We continue the study of counting complexity begun in [Buergisser, Cucker 04] and [Buergisser, Cucker, Lotz 05] by proving upper and lower bounds on the complexity of computing the Hilbert polynomial of  ...  We thank Felipe Cucker for discussions and inviting us to Hong Kong in Spring 2004, where the basis of this work was elaborated in the joint papers [14, 13] .  ... 
arXiv:cs/0502044v1 fatcat:suvi4npdivct3ginrqbvgkk7ra

Schur Polynomials do not have small formulas if the Determinant doesn't! [article]

Prasad Chaugule, Mrinal Kumar, Nutan Limaye, Chandra Kanta Mohapatra, Adrian She, Srikanth Srinivasan
2019 arXiv   pre-print
To the best of our knowledge, these are one of the first hardness results of this kind for families of polynomials which are not multilinear.  ...  They play a central role in the study of Symmetric functions, in Representation theory [Sta99], in Schubert calculus [LM10] as well as in Enumerative combinatorics [Gas96, Sta84, Sta99].  ...  The authors thank Amir for allowing us to include Question 4.18 and related discussion in this draft.  ... 
arXiv:1911.12520v1 fatcat:rpgk2xt6lbar5bg6irfggzlleq

Algorithm 795: PHCpack: a general-purpose solver for polynomial systems by homotopy continuation

Jan Verschelde
1999 ACM Transactions on Mathematical Software  
The outline of one black-box solver is sketched, and a report is given on its performance on a large database of test problems.  ...  Polynomial systems occur in a wide variety of application domains.  ...  The computation of mixed volumes is a crucial step in the resolution of sparse polynomial systems.  ... 
doi:10.1145/317275.317286 fatcat:33xrvqlxszc3vowcyc4uhr72uu

High, low, and quantitative roads in linear algebra

Robert C. Thompson
1992 Linear Algebra and its Applications  
Exact integer and multivariable symbolic computations are possible in Derive, Macsyma, Mathematics, Maple, Milo, and Theorist, permitting the testing of conjectures with a ring theory structure.  ...  A future release will include sparse matrix algorithms.  ...  The algebra of Schubert cycles is a homomorphic image of the algebra of symmetric polynomials.  ... 
doi:10.1016/0024-3795(92)90371-g fatcat:k7pu6jiwxnha3hs6x6zhwz5bim

Newton polytopes and numerical algebraic geometry [article]

Taylor Brysiewicz
2020 arXiv   pre-print
Based on recent results of Esterov, we give an algorithm which recursively solves a sparse polynomial system when the support of that system is either lacunary or triangular.  ...  The first algorithm we develop functions as a numerical oracle for the Newton polytope of a hypersurface and is based on ideas of Hauenstein and Sottile.  ...  Sparse polynomial systems belonging to the same family share a striking number of properties, many depending only on the collection P • of convex hulls of the supports in A • , called Newton polytopes.  ... 
arXiv:2004.12177v1 fatcat:mfkodyanyrhctorz5aseri7j7y

Advanced determinant calculus: A complement

C. Krattenthaler
2005 Linear Algebra and its Applications  
Moreover, I add a list of determinant evaluations which I consider as interesting, which have been found since the appearance of the previous article, or which I failed to mention there, including several  ...  In the present article, I share with the reader my experience of applying the methods described in the previous article in order to solve a particular problem from number theory [G. Almkvist, C.  ...  comments and discussions which helped to improve the contents of this paper considerably.  ... 
doi:10.1016/j.laa.2005.06.042 fatcat:dzwf5rvsa5fvhmidbbb5nresyi

Advanced Determinant Calculus: A Complement [article]

Christian Krattenthaler
2005 arXiv   pre-print
Moreover, I add a list of determinant evaluations which I consider as interesting, which have been found since the appearance of the previous article, or which I failed to mention there, including several  ...  In the present article, I share with the reader my experience of applying the methods described in the previous article in order to solve a particular problem from number theory (G. Almkvist, J.  ...  Moreover, I am extremely grateful to Dave Saunders and Zhendong Wan who performed the LinBox computations of determinants of size 3840, without which it would have been impossible for me to formulate Conjectures  ... 
arXiv:math/0503507v2 fatcat:dhwgihkabfgzjefarob2m5zcvm

Computing quantum dynamics in the semiclassical regime [article]

Caroline Lasser, Christian Lubich
2020 arXiv   pre-print
to the direct computation of expectation values of observables.  ...  Making good use of classical mechanics is essential for all these approaches. The arising aspects of time integration and high-dimensional quadrature are also discussed.  ...  large m: denoting by I Γ the interpolation operator on a sparse grid Γ in d dimensions that corresponds to a hyperbolic cross for the Fourier coefficients (see, e.g.  ... 
arXiv:2002.00624v2 fatcat:wtot6wuwvzfjrc7lxccyxroq4u
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