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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, Tensor

Portions 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.

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Syzygy

Cite this as:

Germundsson, Roger and Weisstein, Eric W. "Syzygy." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Syzygy.html

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