DefinitionDefinition: Literature-sourcedProof: Not applicableaudited 2026-09-06

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

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homogeneous polynomial and homogeneous ideal

Definition

Put $S = k [x_{0}, \dots, x_{n}]$ with total-degree grading. A polynomial is homogeneous of degree $d$ if each occurring monomial has total degree $d$ ; $0$ is homogeneous in every degree. An ideal $J \subseteq S$ is homogeneous if $F = \sum_{d} F_{d} \in J$ implies $F_{d} \in J$ for every $d$ .

Depends on

Used by

affine cone projective set Definition
degree projective hypersurface Definition
homogeneous ideal saturation Definition
projective algebraic set Definition
homogeneous polynomial zero locus well defined Lemma
homogenization dehomogenization correspondence Lemma
projective irreducibility homogeneous prime Lemma

Dependency tree · two levels

4 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

J. S. Milne, Algebraic Geometry, Chapter 6 (standard reference, not scraped)
Michael Artin, Algebraic Geometry, Chapter 3 (standard reference, not scraped)