Built on

B. Buchberger, “An Algorithmical Criterion for the Solvability of Algebraic Systems of Equations,” Aequationes Mathematicae
1970
Earlier work this paper cites.
H. Möller, “On decomposing systems of polynomial equations with finitely many solutions,” Applicable Algebra in Engineering, Communication and Computing
1993
Earlier work this paper cites.
Z. Bern, L. J. Dixon, D. C. Dunbar, and D. A. Kosower, “One-Loop n-Point Gauge Theory Amplitudes, Unitarity and Collinear Limits,” Nucl. Phys
1994
Earlier work this paper cites.
Springer, 2th ed., 1997
D. A. Cox, J. B. Little, and D. O’Shea, Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra · 1997
Earlier work this paper cites.
F. Rouillier, “Solving zero-dimensional systems through the rational univariate representation,” Applicable Algebra in Engineering, Communication and Computing
1999
Earlier work this paper cites.
AMS, 2002
B. Sturmfels, Solving Systems of Polynomial Equations · 2002
Earlier work this paper cites.
F. Cachazo, P. Svrcek, and E. Witten, “MHV vertices and tree amplitudes in gauge theory,” JHEP
2004
Earlier work this paper cites.
R. Britto, F. Cachazo, and B. Feng, “Generalized unitarity and one-loop amplitudes in N = 4 super-Yang-Mills,” Nucl. Phys
2005
Earlier work this paper cites.
R. Britto, F. Cachazo, and B. Feng, “New Recursion Relations for Tree Amplitudes of Gluons,” Nucl. Phys
2005
Earlier work this paper cites.
Springer, 2th ed., 2005
D. A. Cox, J. B. Little, and D. O’Shea, Using Algebraic Geometry · 2005
Earlier work this paper cites.
G. Ossola, C. G. Papadopoulos, and R. Pittau, “Reducing full one-loop amplitudes to scalar integrals at the integrand level,” Nucl.Phys
2007
Earlier work this paper cites.

Similar

Z. Bern, M. Czakon, L. J. Dixon, D. A. Kosower, and V. A. Smirnov, “The Four-Loop Planar Amplitude and Cusp Anomalous Dimension in Maximally Supersymmetric Yang-Mills Theory,” Phys.Rev
2007
Cited alongside, same era.
G. Ossola, C. G. Papadopoulos, and R. Pittau, “Numerical evaluation of six-photon amplitudes,” JHEP
Original
2007
Cited alongside, same era.
C. Berger, Z. Bern, L. Dixon, F. Febres Cordero, D. Forde, et al
Original
2008
Cited alongside, same era.
W. Giele and G. Zanderighi, “On the Numerical Evaluation of One-Loop Amplitudes: The Gluonic Case,” JHEP
Original
2008
Cited alongside, same era.
R. K. Ellis, W. T. Giele, and Z. Kunszt, “A Numerical Unitarity Formalism for Evaluating One-Loop Amplitudes,” JHEP

Then

P. Mastrolia, G. Ossola, T. Reiter, and F. Tramontano, “Scattering AMplitudes from Unitarity-based Reduction Algorithm at the Integrand-level,” JHEP
Original
2010
Later among the works it cites.
S. Badger, B. Biedermann, and P. Uwer, “NGluon: A Package to Calculate One-loop Multi-gluon Amplitudes,” Comput.Phys.Commun
Original
2011
Later among the works it cites.
V. Hirschi, R. Frederix, S. Frixione, M. V. Garzelli, F. Maltoni, et al
Original
2011
Later among the works it cites.
P. Mastrolia and G. Ossola, “On the Integrand-Reduction Method for Two-Loop Scattering Amplitudes,” JHEP
Original
2011
Later among the works it cites.
J. Gluza, K. Kajda, and D. A. Kosower, “Towards a Basis for Planar Two-Loop Integrals,” Phys.Rev

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2012

Scattering Amplitudes from Multivariate Polynomial Division

Mastrolia, Pierpaolo, Mirabella, Edoardo, Ossola, Giovanni et al.

Understand

We show that the evaluation of scattering amplitudes can be formulated as a problem of multivariate polynomial division, with the components of the integration-momenta as indeterminates.

We present a recurrence relation which, independently of the number of loops, leads to the multi-particle pole decomposition of the integrands of the scattering amplitudes.
The recursive algorithm is based on the Weak Nullstellensatz Theorem and on the division modulo the Groebner basis associated to all possible multi-particle cuts.
We apply it to dimensionally regulated one-loop amplitudes, recovering the well-known integrand-decomposition formula.

Built on

B. Buchberger, “An Algorithmical Criterion for the Solvability of Algebraic Systems of Equations,” Aequationes Mathematicae
1970
Earlier work this paper cites.
H. Möller, “On decomposing systems of polynomial equations with finitely many solutions,” Applicable Algebra in Engineering, Communication and Computing
1993
Earlier work this paper cites.
Z. Bern, L. J. Dixon, D. C. Dunbar, and D. A. Kosower, “One-Loop n-Point Gauge Theory Amplitudes, Unitarity and Collinear Limits,” Nucl. Phys
1994
Earlier work this paper cites.
Springer, 2th ed., 1997
D. A. Cox, J. B. Little, and D. O’Shea, Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra · 1997
Earlier work this paper cites.
F. Rouillier, “Solving zero-dimensional systems through the rational univariate representation,” Applicable Algebra in Engineering, Communication and Computing
1999
Earlier work this paper cites.
AMS, 2002
B. Sturmfels, Solving Systems of Polynomial Equations · 2002
Earlier work this paper cites.
F. Cachazo, P. Svrcek, and E. Witten, “MHV vertices and tree amplitudes in gauge theory,” JHEP
2004
Earlier work this paper cites.
R. Britto, F. Cachazo, and B. Feng, “Generalized unitarity and one-loop amplitudes in N = 4 super-Yang-Mills,” Nucl. Phys
2005
Earlier work this paper cites.
R. Britto, F. Cachazo, and B. Feng, “New Recursion Relations for Tree Amplitudes of Gluons,” Nucl. Phys
2005
Earlier work this paper cites.
Springer, 2th ed., 2005
D. A. Cox, J. B. Little, and D. O’Shea, Using Algebraic Geometry · 2005
Earlier work this paper cites.
G. Ossola, C. G. Papadopoulos, and R. Pittau, “Reducing full one-loop amplitudes to scalar integrals at the integrand level,” Nucl.Phys
2007
Earlier work this paper cites.