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Quantum Integer Programming (QuIP) 47-779: Lecture Notes [article]

David E. Bernal, Sridhar Tayur, Davide Venturelli
2021 arXiv   pre-print
of near term quantum and quantum-inspired computing in solving optimization problems.  ...  Graver Augmented Multiseed Algorithm (GAMA), Simulated or Quantum Annealing and QAOA, and ideas on how to solve these Integer problems using these quantum and classical methods.  ...  Hence, if we find a way to solve IP in polynomial time, we can solve all NP problems in polynomial time and prove the result P=NP!  ... 
arXiv:2012.11382v2 fatcat:kin4pzosivc53d5sf6qua4mu2m

Book reports

2005 Computers and Mathematics with Applications  
Taylor Formula and Grobner Duality. 32. Grobner III. 32.1. Macaulay Bases. 32.2. Macaulay Basis and Grobner Representation. 32.3. Macaulay Basis and Decomposition of Primary Ideals. 32.4.  ...  Multiplicity of Primary Ideals. 30.7. The Structure of Primary Ideals at the Origin. 31. Grobner II. 31.1. Noetherian Equations. 31.2. Stability. 31.3. Grobner Duality. 31.4. Leibniz Formula. 31.5.  ... 
doi:10.1016/j.camwa.2005.10.001 fatcat:smjkjmzbkfdbvezerczxaktjfy

Minimizing Polynomial Functions [article]

Pablo A. Parrilo, Bernd Sturmfels
2001 arXiv   pre-print
We compare algorithms for global optimization of polynomial functions in many variables.  ...  This paper was presented at the Workshop on Algorithmic and Quantitative Aspects of Real Algebraic Geometry in Mathematics and Computer Science, held at DIMACS, Rutgers University, March 12-16, 2001.  ...  Gröbner bases and eigenvalues. We review the method of solving polynomial equations by means of Gröbner bases and eigenvalues [CLO98, §2.4] .  ... 
arXiv:math/0103170v1 fatcat:fs43ly7vsbe6zmaqo2psxmstoq

Recent advances in computational methods for the power flow equations

Dhagash Mehta, Daniel K. Molzahn, Konstantin Turitsyn
2016 2016 American Control Conference (ACC)  
The power flow equations are a system of multivariate nonlinear equations which relate the power injections and voltages in a power system.  ...  After briefly reviewing some of the traditional computational methods used to solve the power flow equations, we focus on three emerging methods: the numerical polynomial homotopy continuation method,  ...  THE GRÖBNER BASIS TECHNIQUES As discussed in the previous sections, many important power systems problems are naturally formulated in terms of systems of polynomial equations.  ... 
doi:10.1109/acc.2016.7525170 dblp:conf/amcc/MehtaMT16 fatcat:ravboieb7rcuxj3db25ksk4gyu


2013 Glasgow Mathematical Journal  
We present a unifying framework for the key concepts and results of higher Koszul duality theory for N-homogeneous algebras: the Koszul complex, the candidate for the space of syzygies and the higher operations  ...  For that we introduce a general notion: Gröbner bases for algebras over non-symmetric operads.  ...  By Theorem 3.1, it is enough to prove that all S-polynomials coming from small common multiples of leading terms of the given relations can be reduced to zero using the same system of relations.  ... 
doi:10.1017/s0017089513000505 fatcat:tt2cvrugifhvjmmiruwhsdoq7i

Polynomial and rational solutions of holonomic systems [article]

T. Oaku, N. Takayama, H. Tsai
2000 arXiv   pre-print
The aim of this paper is to give two new algorithms, which are elimination free, to find polynomial and rational solutions for a given holonomic system associated to a set of linear differential operators  ...  in the Weyl algebra D = k where k is a subfield of the complex numbers.  ...  To the authors, solving holonomic systems is analogous to solving systems of algebraic equations of zero-dimensional ideals.  ... 
arXiv:math/0001064v1 fatcat:5kv24vvmabdlfhxv2zc5zcb65e

Globally Optimal Solution to Inverse Kinematics of 7DOF Serial Manipulator [article]

Pavel Trutman
2020 arXiv   pre-print
We demonstrate the method on the 7DOF KUKA LBR IIWA manipulator and show that we are able to compute the optimal IK or certify in-feasibility in 99 % tested poses.  ...  This is important since it signicantly simplies further step where we nd the optimal solution by Lasserre relaxations of non-convex polynomial systems.  ...  It leads to solving a polynomial system with 16 solutions.  ... 
arXiv:2007.12550v1 fatcat:kh6bkj36fbdd7m4hfb2wlfthvm

On Rational Interpolation-Based List-Decoding and List-Decoding Binary Goppa Codes

Peter Beelen, Tom Hoholdt, Johan S. R. Nielsen, Yingquan Wu
2013 IEEE Transactions on Information Theory  
Finally, we point out a connection between the governing equations of the Wu algorithm and the Guruswami-Sudan algorithm (GSA), immediately leading to equality in the decoding range and a duality in the  ...  choice of parameters needed for decoding, both in the case of GRS codes and in the case of Goppa codes.  ...  of Algebraic Geometry in Coding Theory and Cryptography.  ... 
doi:10.1109/tit.2013.2243800 fatcat:rduksenzaza4lpup4gqqwed3t4

Scattering equations and global duality of residues

Mads Søgaard, Yang Zhang
2016 Physical Review D  
We explain how the Bezoutian matrix facilitates the calculation of amplitudes in the CHY formalism, without explicitly solving the scattering equations or summing over the individual residues.  ...  We examine the polynomial form of the scattering equations by means of computational algebraic geometry.  ...  Here we follow Dolan and Goddard [11] , who recently proved that the n-point scattering equations, after properly fixing the SL(2, C) redundancy, are equivalent to a system of (n − 3) polynomial equations  ... 
doi:10.1103/physrevd.93.105009 fatcat:kpm5xgvi7beyjiymhzvf5ff3ya

Page 3137 of Mathematical Reviews Vol. , Issue 2002E [page]

2002 Mathematical Reviews  
The polynomials defin- ing a K-algebra A as a quotient of a free K-algebra are combined with the polynomials defining a one-sided ideal J of A by using a tagging notation in order to forbid left multiplication  ...  Thus the algorithms developed rely on earlier re- search devoted to Grobner basis theory and presented in several publications by Buchberger, Mora, Reinert, Ufnarovskii, Winkler and other authors.  ... 

Improvement on the vanishing component analysis by grouping strategy

Xiaofeng Zhang
2018 EURASIP Journal on Wireless Communications and Networking  
, and solves the classification problem in ideal space dual to kernel space.  ...  The classification decision function did not use a predetermined threshold; instead, it solved the values of all polynomials of vanishing component and sorted them, and then used majority voting approach  ...  Kiral et al. raised two dualities, namely, the duality between kernel and ideal and the duality between ideal and manifold pattern.  ... 
doi:10.1186/s13638-018-1112-7 fatcat:x5bcwny565gmxapgtbebvkjgsu

Computing Multidimensional Residues [article]

Eduardo Cattani, Alicia Dickenstein, Bernd Sturmfels
1994 arXiv   pre-print
All algorithms are illustrated for an explicit system in three variables.  ...  The global residue of a polynomial equals the highest coefficient of its (Groebner basis) normal form, and, conversely, the entire normal form is expressed in terms of global residues.  ...  Dickenstein was partially supported by UBACYT and CONICET, and B. Sturmfels was partially supported by NSF grants DMS-9201453, DMS-9258547 (NYI) and a David and Lucile Packard Fellowship.  ... 
arXiv:alg-geom/9404011v1 fatcat:pti6kxc7yrdcng4c4xo6przira

An Improved Semidefinite Programming Hierarchy for Testing Entanglement

Aram W. Harrow, Anand Natarajan, Xiaodi Wu
2017 Communications in Mathematical Physics  
This yields an algorithm for separability testing which is singly exponential in dimension and polylogarithmic in accuracy.  ...  Our analysis makes use of tools from algebraic geometry, but our algorithm is elementary and differs from DPS only by one simple additional collection of constraints.  ...  All three authors (AWH, AN and XW) were funded by ARO contract W911NF-12-1-0486.  ... 
doi:10.1007/s00220-017-2859-0 fatcat:zrnui23r6vaf3ofduwykg33hai

The distance to cubic symmetry class as a polynomial optimization problem [article]

Perla Azzi
2022 arXiv   pre-print
using the technique of Gröbner bases.  ...  We propose to make use of this mathematical property to formulate the distance to cubic symmetry problem as a polynomial (in fact quadratic) optimization problem, and to derive its quasi-analytical solution  ...  Solving algebraic systems using Gröbner bases In this appendix, we propose to explain how to use Gröbner bases to solve non-linear algebraic systems.  ... 
arXiv:2203.14562v1 fatcat:w3zcrp67rzgqjpzfub4kgthzie

Computing multidimensional residues [chapter]

E. Cattani, A. Dickenstein, B. Sturmfels
1996 Algorithms in Algebraic Geometry and Applications  
Given n polynomials in n variables with a finite number of complex roots, for any of their roots there is a local residue operator assigning a complex number to any polynomial.  ...  In this paper we present symbolic algorithms for evaluating that rational function.  ...  Dickenstein was partially supported by UBACYT and CONICET, and B. Sturmfels was partially supported by NSF grants DMS-9201453, DMS-9258547 (NYI) and a David and Lucile Packard Fellowship.  ... 
doi:10.1007/978-3-0348-9104-2_8 fatcat:t2bqhxfxy5bvppclhifxqwwfnm
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