Syzygy
A technical mathematical object defined in terms of a polynomial ring of n variables over a field k. 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
| epsilon_(ijk)delta_(lm)-epsilon_(jkl)delta_(im)+epsilon_(kli)delta_(jm)-epsilon_(lij)delta_(km)=0, |
where epsilon_(ijk) is the permutation tensor and delta_(ij) is the Kronecker delta.
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
| q_1f_1+q_2f_2+...+q_mf_m=g. |
Syzygies give the q_i 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.
See also
Fundamental System, Hilbert Basis Theorem, Isotropic Tensor, Kronecker Delta, Syzygies Problem, TensorPortions of this entry contributed by Roger Germundsson
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References
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. London 143, 407-548, 1853.Referenced on Wolfram|Alpha
SyzygyCite this as:
Weisstein, Eric W., with contributions by Roger Germundsson. "Syzygy." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Syzygy.html