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Eisenbud-Green-Harris Conjecture

The Eisenbud-Green-Harris conjecture asserts that a homogeneous ideal I in the polynomial ring S=k[x_1,...,x_n] over a field k, containing a ring regular sequence of homogeneous polynomials of total degrees 2<=a_1<=...<=a_c has a lex-plus-powers ideal J for these powers with

H(S/I,d)=H(S/J,d).

This equality is required in every integer total degree d, and H denotes the Hilbert function. The ideal J must be one homogeneous ideal across all degrees, rather than independent choices of monomials in each degree.

See also

Hilbert Function, Homogeneous Ideal, Lex-Plus-Powers Conjecture, Lex-Plus-Powers Ideal, Ring Regular Sequence

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References

Caviglia, G.; De Stefani, A.; and Sbarra, E. "The Eisenbud-Green-Harris Conjecture." In Commutative Algebra (Ed. I. Peeva). Cham, Switzerland: Springer, pp. 159-187, 2021. https://doi.org/10.1007/978-3-030-89694-2_5.

Cite this as:

Weisstein, Eric W. "Eisenbud-Green-Harris Conjecture." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Eisenbud-Green-HarrisConjecture.html

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