A technical mathematical object defined in terms of a polynomial ring of
variables over a field. Syzygies occur in tensors at rank
5, 7, 8, and all higher ranks, and play a role in restricting the number of independent
isotropic tensors. An example of a rank-5 syzygy
is
Syzygies can roughly be viewed as an extension of polynomial greatest common divisors to the multivariable case, i.e., they give a method for solving multivariate polynomial Diophantine equations
Syzygies give the
polynomials or else show that no such solution exists. The ability to solve linear
multivariable polynomial equations allows computation of multivariate ideal operations
such intersection, quotient, and a number of other commutative algebra operations.
Hilbert, D. "Über die Theorie der algebraischen Formen." Math. Ann.36, 473-534, 1890.Iyanaga, S.
and Kawada, Y. (Eds.). "Syzygy Theory." §364F in Encyclopedic
Dictionary of Mathematics. Cambridge, MA: MIT Press, p. 1140, 1980.Olver,
P. J. "Syzygies." Classical
Invariant Theory. Cambridge, England: Cambridge University Press, pp. 110-112,
1999.Sylvester, J. J. "On a Theory of Syzygetic Relations
of Two Rational Integral Functions, Comprising an Application of the Theory of Sturm's
Functions, and that of the Greatest Algebraic Common Measure." Philos. Trans.
Roy. Soc. London143, 407-548, 1853.
variables over a field 
. Syzygies occur in tensors at rank
5, 7, 8, and all higher ranks, and play a role in restricting the number of independent
isotropic tensors. An example of a rank-5 syzygy
is
is the permutation
tensor and 
is the Kronecker delta.
polynomials or else show that no such solution exists. The ability to solve linear
multivariable polynomial equations allows computation of multivariate ideal operations
such intersection, quotient, and a number of other commutative algebra operations.
