In the following we will define gradings onk[x1, . . . , xn]. These definitions can easily be generalized to any semigroup ringR[S], but we stick to the polynomial ring to keep things simple.

Definition 5.3.1 By a grading on k[x_{1}, . . . , xn] we mean a monoid homomor-
phismφ :N^{n} → S to a semigroup S. Foru ∈ N^{n} the φ-degree of x^{u} and u is
φ(u).

Definition 5.3.2 Let φ be a grading on k[x_{1}, . . . , x_{n}]. A polynomial f ∈
k[x1, . . . , xn] is φ-homogeneous if every exponent of f has the same φ-degree.

An ideal I ⊆ k[x_{1}, . . . , xn] is φ-homogeneous if it has a generating set of φ-
homogeneous polynomials.

Example 5.3.3 Thestandard grading onk[x_{1}, . . . , xn] is defined asφ:N^{n}→N
with φ(u) = Pn

i=1u_{i} for u ∈ N^{n}. We sometimes call φ(u) the total degree of
x^{u}. A polynomial is homogeneous in the standard grading if every term has the
same usual total degree.

• x^{3}_{1}−x^{2}_{2}x3+x^{3}_{3} is homogeneous in the standard grading.

• x^{2}_{1}−x^{1}_{2} is not homogeneous in the standard grading.

• hx_{1} −x_{2}, x^{3}_{1} −x^{2}_{2}x_{3} +x^{3}_{3} +x_{1} −x_{2}i = hx_{1} −x_{2}, x^{3}_{1} −x^{3}_{2}x_{3} +x^{3}_{3}i is a
homogeneous ideal in the standard grading.

Example 5.3.4 Let ω ∈ R^{n}. We may define the grading φω : N^{n} → R by
φ_{ω}(u) := ω·u. For a polynomial f ∈ k[x_{1}, . . . , x_{n}] the initial form in_{ω}(f) is
φω-homogeneous by Definition 1.4.1. For simplicity we sometimes just say ω-
homogeneous. If I ⊆ k[x_{1}, . . . , xn] is an ideal then the initial ideal inω(I) is
ω-homogeneous by Definition 1.6.1.

Example 5.3.5 A matrix A ∈ N^{d×n} defines a grading φA : N^{n} → N^{d} by
φ_{A}(u) := Au. Notice that if ω is a a vector in the rowspace of A then any
φ-homogeneous polynomial is also ω-homogeneous.

A priori, it is not clear that the last ideal in Example 5.3.3 is homogeneous.

In the following we will find an algorithm for deciding if an ideal is homogeneous.

Lemma 5.3.6 Let φ be a grading on k[x_{1}, . . . , xn]. Let f, g ∈ k[x_{1}, . . . , xn]
be φ-homogeneous and h ∈ k[x_{1}, . . . , xn] a single term. The polynomial hf is
φ-homogeneous. If the terms of f and g have the same φ-degree, then f +g is
φ-homogeneous.

Proof. Leth =cx^{u} and let c^{′}x^{u}^{′} and c^{′′}x^{u}^{′′} be two terms off resulting in two
termscc^{′}x^{u+u}^{′} and cc^{′′}x^{u+u}^{′′}. We check that they have the same φ-degree:

φ(u+u^{′}) =φ(u) +φ(u^{′}) =φ(u) +φ(u^{′′}) =φ(u+u^{′′}).

Hencehf isφ-homogeneous. It is clear from the definition thatf+gis homo- geneous since all terms have the sameφ-degree. 2

Lemma 5.3.7 Let f, g ∈k[x1, . . . , xn]\ {0} be φ-homogeneous and ≺ a term ordering. Then the S-polynomialS≺(f, g) is φ-homogeneous.

Proof. The S-polynomial was defined as
S_{≺}(f, g) = lcm(in_{≺}(f),in_{≺}(g))

in_{≺}(f) f −lcm(in_{≺}(f),in_{≺}(g))
in_{≺}(g) g

to carefully make two terms cancel - one from each summand. Each summand isφ-homogeneous by Lemma 5.3.6. Since two terms cancel they must have the sameφ-degree. Therefore all terms of the S-polynomial have the sameφ-degree.

2

Lemma 5.3.8 Let f and f1, . . . , fs be φ-homogeneous polynomials. The re- mainderrproduced by the division algorithm (Algorithm 1.5.1) isφ-homogeneous.

Proof. In the division algorithm f is assigned to p and p is adjusted until it eventually becomes zero and the algorithm terminates. At the beginning p is φ-homogeneous because f is. In each iteration the p remains φ-homogeneous because we only subtract φ-homogeneous polynomials from p of the same φ- degree. Thereforepremainsφ-homogeneous of the same degree until it becomes 0. Terms are moved fromp to the remainderr. Therefore ther = 0 when the algorithm terminates. 2

Proposition 5.3.9 Let φ be a grading on k[x_{1}, . . . , xn]. Let I ⊆k[x_{1}, . . . , xn]
be a φ-homogeneous ideal and ≺ a term ordering. Then the reduced Gr¨obner
basis G≺(I) is φ-homogeneous.

Proof. By Hilbert’s Basis Theorem 1.1.6 there exists a finite generating set
G⊆k[x_{1}, . . . , x_{n}] for I. We also know that there exists a generating set G^{′} ⊆
k[x1, . . . , xn] of φ-homogeneous polynomials since I is φ-homogeneous, which
could be infinite. We now take everyg∈Gand express it in terms of elements
ofG^{′}:

g=

m

X

i=1

figi

for somem ∈N,fi ∈k[x_{1}, . . . , xn] andgi ∈G^{′}. Doing so for all finitely many
terms in g requires only a finite number of terms of the type g^{′}_{i}. We let G^{′′}

denote this finite set ofφ-homogeneous polynomials which generate I.

If we perform Buchberger’s Algorithm on G^{′′}, the result is φ-homogeneous
because the operations of taking S-polynomials and remainder preserves ho-
mogeneity (Lemma 5.3.7 and Lemma 5.3.8. The minimizing and autoreducing
algorithms (Algorithm 1.7.8 and Algorithm 1.7.9) also preserveφ-homogeneity
and together produce the unique reduced Gr¨obner basis ofI with respect to≺.
We conclude that this Gr¨obner basis isφ-homogeneous. 2

Knowing that the reduced Gr¨obner basis is always homogeneous can be used to compute a homogeneous generating set for a homogeneous ideal but also to check if and ideal is homogeneous.

Algorithm 5.3.10

Input: A set G⊆k[x_{1}, . . . , xn]and a grading φ.

Output: “Yes” if the ideal hGi is φ-homogeneous and “No” otherwise.

• Compute the reduced Gr¨obner basis H:=G≺(hGi).

• Return “Yes” if all polynomials in H are φ-homogeneous and “No” oth- erwise.

Example 5.3.11 The ideal hx^{2} +y −3zx, y^{2}z +zx−1i ⊆ Q[x, y, z] is not
homogeneous in the standard grading because its reduced Gr¨obner basis with
respect to the lexicographic ordering (withz≺y≺x) is not homogeneous:

{y^{4}z^{2}+ 1−3z^{2}+yz^{2}−2y^{2}z+ 3y^{2}z^{3}, x−3z+yz−y^{2}+ 3y^{2}z^{2}+y^{4}z}.
Definition 5.3.12 By aφ-homogeneous part of a polynomial f we mean the
sum of all terms inf of a particularφ-degree.

Lemma 5.3.13 Let φ be a grading on R := k[x_{1}, . . . , x_{n}]. Let I ⊆ R be a
φ-homogeneous ideal. A polynomial f ∈ I if and only if every φ-homogeneous
part of f is inI.

Proof. Clearly, if every φ-homogeneous part of f is in I then so is the sum, which equals f. On the other hand, let f ∈ I be a polynomial, let h be a φ-homogeneous part of f, and let G≺(I) be the reduced Gr¨obner basis of I with respect to some term ordering ≺. The division algorithm will produce an expression f = P

iaifi with a_{1} being polynomials and fi ∈ G≺(I). By
splitting each a_{i} into terms and multiplying out we get an expression f =
P

jbjgj wherebj is a single term andgj ∈ G≺(I). Since eachbj is homogeneous, any homogeneous part of f is written as f = P

j∈Jbjgj for J chosen to give
just the terms in the right φ-degree. Since g_{j} ∈ I we conclude that the φ-
homogeneous part is inI. 2

Proposition 5.3.14 Let R := k[x_{1}, . . . , x_{n}] and φ :N^{n} → S a grading. As a
k-vector space we may write R as a direct sum:

R= M

m∈S

R_{m}

where Rm is thek-vector space of homogeneous polynomials of φ-degree m (to- gether with 0). Furthermore, if I is a φ-homogeneous ideal then we can define Im to be the set of φ-homogeneous polynomials in I of degree m (together with 0). As a vector space we have

I = M

m∈S

Im.

Proof. Clearly, a polynomial can be uniquely be split into finitely many non- zero homogeneous parts of differentφ-degree, which provesR=L

m∈SRm. A polynomial f ∈ I is also in R and therefore splits into φ-homogeneous parts.

Each of these is inI by Corollary 5.3.13 and therefore in an I_{m}. 2