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Homogeneous Ideal

A homogeneous ideal I in a graded ring R= direct sum A_i is an ideal generated by a set of homogeneous elements, i.e., each one belongs to some A_i. For example, the polynomial ring C[x]= direct sum A_i is a graded ring, where A_i={ax^i}. The ideal I=<x^2>, i.e., all polynomials with no constant or linear terms, is a homogeneous ideal in C[x]. Another homogeneous ideal is I=<x^2+y^2+z^2,xy+yz+zx,z^5> in C[x,y,z].

Given any finite set of polynomials in n variables, the process of homogenization converts them to homogeneous polynomials in n+1 variables. If f=f(x_1,...,x_n) is a polynomial of degree d then

f^h(x_0,x_1,...,x_n)=x_0^df(x_1/x_0,...,x_n/x_0)

is the homogenization of f. Similarly, if I is an ideal in C[x_1,...,x_n], then I^h=<f^h:f in I> is its homogenization and is a homogeneous ideal. For example, if f=x_1^3+2x_1x_2-3 then f^h=x_1^3+2x_0x_1x_2-3x_0^3. Note that in general, if I=<f_1,...,f_k> then I^h may have more elements than <f_1^h,...,f_k^h>. However, if f_1, ..., f_k form a Gröbner basis using a graded monomial order, then I^h=<f_1^h,...,f_k^h>. A polynomial is easily dehomogenized by setting the extra variable x_0=1.

The affine variety V corresponding to a homogeneous ideal has the property that x in V iff cx in V for all complex numbers c. Therefore, a homogeneous ideal defines an algebraic variety in complex projective space.

See also

Algebraic Variety, Category Theory, Commutative Algebra, Conic Section, Eisenbud-Green-Harris Conjecture, Ideal, Lex-Plus-Powers Ideal, Monomial Ideal, Prime Ideal, Scheme, Zariski Topology

Portions of this entry contributed by Todd Rowland

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Cite this as:

Weisstein, Eric W., with contributions by Todd Rowland. "Homogeneous Ideal." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/HomogeneousIdeal.html

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