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A technical mathematical object defined in terms of a Polynomial Ring of 
variables over a Field 
.
See also Fundamental System, Hilbert Basis Theorem, Syzygies Problem
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.