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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 is contained in only one of the 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 c. Therefore, a homogeneous ideal defines an algebraic variety in complex projective space.

See also

Algebraic Variety, Category Theory, Commutative Algebra, Conic Section, Ideal, Prime Ideal, Projective Algebraic Variety, Scheme, Zariski Topology

This entry contributed by Todd Rowland

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Cite this as:

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

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