Engineering · Mathematics · Advanced Algebra Handbook
Multivariable Division and Polynomial Ideal Reduction Bases
Commutative algebra studies ideals, prime structure, factorisation, localisation, algebraic dependence and dimension. It is most reliable when global questions are reduced to ideal-theoretic or local statements and then reassembled. This page consolidates the relevant material from the supplied algebra source into a stand-alone handbook chapter.
Executive summary
This chapter develops multivariable division and polynomial ideal reduction bases as part of a connected advanced-algebra learning sequence. The emphasis is on definitions, hypotheses, structural results and repeatable methods rather than historical narrative.
The source material is theorem-rich. Accordingly, the handbook presentation separates vocabulary from results and then adds a verification workflow so that each statement can be applied safely. Mathematical examples in the source are treated as examples, not as universal rules.
Problem-solving workflow
Identify the ring, its units and its relevant ideals.
Check finiteness or chain conditions before using finite-generation arguments.
Use prime or maximal ideals to convert multiplicative questions into quotient-domain or quotient-field questions.
Localise when the property can be tested near a prime or maximal ideal.
Track integrality, dimension or ideal factorisation with the exact hypotheses stated.
Return from local or quotient data to the original ring only through a justified correspondence.
Core definitions
Definition
A monomial order is a well-ordering of N n such that α ⪯β implies α + γ ⪯β + γ for all α, β, γ ∈N n. A monomial order will be used as follows. If X = (x1, . . .
Definition
The lexicographic order on N n is defined by α ⪯lex β if either α = β or the first nonzero coordinate in β −α is positive.17 The term lexicographic refers to the standard ordering of words in a dictionary. For example, the following German words are increasing in lexicographic order (the letters are ordered a < b < c < · · · < z): ausgehen ausladen auslagen auslegen bedeuten If α ≺lex β, then they agree for the first i −1 coordinates (for some i ≥1), that is, α1 = β1, . . . , αi−1 = βi−1, and there is strict inequality: αi < βi.
Definition
The degree-lexicographic order on N n is defined by α ⪯dlex β if either α = β or |α| = n i=1 αi < n i=1 βi = |β|, or, if |α| = |β|, then the first nonzero coordinate in β −α is positive. In other words, given α = (α1, . . . , αn) and β = (β1, . . . , βn), first check weights: if |α| < |β|, then α ⪯dlex β; if there is a tie, that is, if α and β have the same weight, then order them lexicographically. For example, (1, 2, 3, 0) ≺dlex (0, 2, 5, 0) and (1, 2, 3, 4) ≺dlex (1, 2, 5, 2).
Definition
Let {g1(X), . . . , gm(X)} be a set of polynomials in k[X]. A polynomial r(X) is reduced mod {g1, . . . , gm} if either r(X) = 0 or no LT(gi) divides any nonzero term of r(X). Here is the division algorithm for polynomials in several variables. Because the algorithm requires the “divisor polynomials” {g1, . . . , gm} to be used in a specific order (after all, an algorithm must give explicit directions), we will be using an m-tuple of polynomials instead of a subset of polynomials. We denote the m-tuple whose ith entry is gi by [g1, . . . , gm], because the usual notation (g1, . . . , gm) would be confused with the ideal (g1, . . . , gm) generated by the gi.
Definition
A set of polynomials {g1, . . . , gm} is a polynomial-reduction basis18 of the ideal I = (g1, . . . , gm) if, for each nonzero f ∈I, there is some gi with LT(gi) | LT( f ). Note that a polynomial-reduction basis is a set of polynomials, not an m-tuple of polynomials. Example 6.128 shows that {y2 + z2, x2y + yz, z3 + xy} is not a polynomial-reduction basis of the ideal (y2 + z2, x2y + yz, z3 + xy).
Definition
Let f (X), g(X) ∈k[X], where LT( f ) = aα Xα and LT(g) = bβ Xβ. Define L( f, g) = Xα∨β. The S-polynomial S( f, g) is defined by S( f, g) = L( f, g) LT( f ) f −L( f, g) LT(g) g; that is, if µ = α ∨β, then S( f, g) = a−1 α Xµ−α f (X) −b−1 β Xµ−βg(X). Note that S( f, g) = −S(g, f ).
Definition
A basis {g1, . . . , gm} of an ideal I is reduced if (i) each gi is monic; (ii) each gi is reduced mod {g1, . . . , gi, . . . , gm}. . . , ft). It can be proved that a reduced polynomial-reduction basis of an ideal is unique. In the special case when each fi(X) is linear, that is, fi(X) = ai1x1 + · · · + ainxn. then the common zeros Var( f1, . . . , ft) are the solutions of a homogeneous system of t equations in n unknowns. Another special case occurs when when f1, . . . , ft are polynomials in one variable. The reduced polynomial-reduction basis obtained from { f1, . . . , ft} turns out to be their gcd, and so the iterated division algorithm has been generalized to polynomials in several variables.
Definition
Let R be a commutative ring. An R-module is an (additive) abelian group M equipped with a scalar multiplication R × M →M, denoted by (r, m) ↦rm, such that the following axioms hold for all m, m′ ∈M and all r,r′, 1 ∈R: (i) r(m + m′) = rm + rm′; (ii) (r + r′)m = rm + r′m; (iii) (rr′)m = r(r′m); (iv) 1m = m.
Principal results and structural facts
Key result
by replacing N with X. ◀ In Lemma 5.70 we constructed, for any set X, a monoid W(X): its elements are the empty word together with all the words xe1 1 · · · xep p on a set X, where p ≥1 and ei = ±1 for all i; its operation is juxtaposition. In contrast to N n, in which all words have length n, the monoid W(X) has words of different lengths. Of more interest here is the submonoid W+(X) of W(X) consisting of all the “positive” words on X: W+(X) = {x1 · · · x p ∈W(X) : xi ∈X and p ≥0}.
Key result
If X is a well-ordered set, then W+(X) is well-ordered in the lexicographic order (which we also denote by ⪯lex).
Key result
shows that α + γ ⪯dlex β + γ . • The next proposition shows, with respect to a monomial order, that polynomials in several variables behave like polynomials in a single variable.
Key result
Let ⪯be a monomial order on N n, and let f (X), g(X), h(X) ∈ k[X] = k[x1, . . . , xn], where k is a field. (i) If Deg( f ) = Deg(g), then LT(g) | LT( f ). (ii) LT(hg) = LT(h)LT(g). (iii) If Deg( f ) = Deg(hg), then LT(g) | LT( f ).
Key result
Let ⪯be a monomial order on N n, and let k[X] = k[x1, . . . , xn]. If f (X) ∈k[X] and G = [g1(X), . . . , gm(X)] is an m-tuple of polynomials in k[X], then there is an algorithm giving polynomials r(X), a1(X), . . ., am(X) ∈k[X] with f = a1g1 + · · · + amgm + r, where r is reduced mod {g1, . . . , gm}, and Deg(aigi) ⪯Deg( f ) for all i.
Key result
A set {g1, . . . , gm} of polynomials is a polynomial-reduction basis of an ideal I = (g1, . . . , gm) if and only if, for each m-tuple Gσ = [gσ(1), . . . , gσ(m)], where σ ∈Sm, every f ∈I has remainder 0 mod Gσ.
Key result
, the division algorithm gives a sequence of reductions h = h0 →h1 → h2 →· · · →h p = 0. But the division algorithm for f adjoins f →h at the front, showing that 0 is the remainder of f mod Gσ, a contradiction. Conversely, assume that every f ∈I has remainder 0 mod Gσ but that {g1, . . . , gm} is not a polynomial-reduction basis of I = (g1, . . . , gm). If there is a nonzero f ∈I with LT(gi) ∤LT( f ) for every i, then in any reduction f gi→ h, we have LT(h) = LT( f ). Hence, if G = [g1, . . . , gm], the division algorithm mod G gives reductions f →h1 →h2 →· · · → h p = r in which LT(r) = LT( f ). Therefore, r ̸= 0; that is, the remainder of f mod G is not zero, and this is a contradiction. •
Key result
Let {g1, . . . , gm} be a polynomial-reduction basis of the ideal I = (g1, . . . , gm), and let G = [g1, . . . , gm]. (i) If f (X) ∈k[X] and Gσ = [gσ(1), . . . , gσ(m)], where σ ∈Sm is a permutation, then the remainder of f mod G is equal to the remainder of f mod Gσ. (ii) A polynomial f ∈I if and only if f has remainder 0 mod G.
Key result
Given g1(X), . . . , gℓ(X) ∈k[X] and monomials c j Xα( j), let h(X) = ℓ j=1 c j Xα( j)g j(X). Let δ be a multidegree. If Deg(h) ≺δ and Deg(c j Xα( j)g j(X)) = δ for all j ≤ℓ, then there are d j ∈k with h(X) = j d j Xδ−µ( j)S(g j, g j+1), where µ( j) = Deg(g j) ∨Deg(g j+1), and for all j < ℓ, Deg ( Xδ−µ( j)S(g j, g j+1) ) ≺δ.
Key result
A set {g1, . . . , gm} is a polynomial-reduction basis of an ideal I = (g1, . . . , gm) if and only if S(gp, gq) has remainder 0 mod G for all p, q, where G = [g1, . . . , gm].
Key result
Every ideal I = ( f1, . . . , fs) in k[X] has a polynomial-reduction basis20 that can be computed by an algorithm.
Key result
(i) If I = ( f1, . . . , ft) is an ideal in k[X], then there is an algorithm to determine whether a polynomial h(X) ∈k[X] lies in I. (ii) If I = ( f1, . . . , ft) ⊆k[X], then there is an algorithm to determine whether a polynomial g(X) ∈k[X] lies in √ I. (iii) If I = ( f1, . . . , ft) and I ′ = ( f ′ 1, . . . , f ′ s) are ideals in k[X], then there is an algorithm to determine whether I = I ′.
Key result
Let k be a field and let k[X] = k[x1, . . . , xn] have a monomial order for which x1 ≻x2 ≻· · · ≻xn (for example, the lexicographic order) and, for fixed p > 1, let Y = x p, . . . , xn. If I ⊆k[X] has a polynomial-reduction basis G = {g1, . . . , gm}, then G ∩IY is a polynomial-reduction basis for the elimination ideal IY = I ∩k[x p, . . . , xn].
Key result
Let k be a field, and let I1, . . . , It be ideals in k[X], where X = x1, . . . , xn. (i) Consider the polynomial ring k[X, y1, . . . , yt] having a new variable y j for each j with 1 ≤j ≤t. If J is the ideal in k[X, y1, . . . , yt] generated by 1 −(y1 + · · · + yt) and y j I j, for all j, then t j=1 I j = JX. (ii) Given polynomial-reduction bases of I1, . . . , It, a polynomial-reduction basis of t j=1 I j can be computed.
Source-grounded examples
Worked source example
Let f (x, y, z) = x2y2 + xy, and let G = [g1, g2, g3], where g1 = y2 + z2 g2 = x2y + yz g3 = z3 + xy. We use the degree-lexicographic order on N 3. Now y2 = LT(g1) | LT( f ) = x2y2, and so f g1 →h, where h = f −x2y2 y2 (y2 + z2) = −x2z2 + xy. The polynomial −x2z2 + xy is reduced mod G, because neither −x2z2 nor xy is divisible by any of the leading terms LT(g1) = y2, LT(g2) = x2y, or LT(g3) = z3. Let us now apply the division algorithm using the 3-tuple G′ = [g2, g1, g3]. The first reduction gives f g2 →h′, where h′ = f −x2y2 x2y (x2y + yz) = −y2z + xy. Now h′ is not reduced, and reducing mod g1 gives h′ −−y2z y2 (y2 + z2) = z3 + xy. But z3 + xy = g3, and so z3 + xy g3 →0. Thus, the remainder depends on the ordering of the divisor polynomials gi in the m-tuple. ◀
Worked source example
Consider the ideal I = (x) ∩(x2, xy, y2) ⊆k[x, y], where k is a field, that we considered in Example 6.117(ii). Even though it is not difficult to find a basis of I by hand, we shall use polynomial-reduction bases to illustrate Proposition 6.140. Let u and v be new variables, and define J = (1 −u −v, ux, vx2, vxy, vy2) ⊆k[x, y, u, v]. The first step is to find a polynomial-reduction basis of J; we use the lex monomial order with x ≺y ≺ u ≺v. It follows from Proposition 6.139 that a polynomial-reduction basis of I is G∩k[x, y]: all those elements of G that do not involve the variables u and v. Thus, I = (x) ∩(x2, xy, y2) = (x2, xy). ◀ We mention that polynomial-reduction bases can be adapted to noncommutative rings.
How to reason with these results
Most advanced-algebra problems become manageable when the representation is separated from the invariant structure. Begin with the definition, then decide whether the problem is asking for an elementwise calculation, a statement about a morphism, or a classification up to isomorphism. That choice determines the correct proof language.
When a theorem gives a structural conclusion, do not jump directly to the conclusion. Write the hypotheses next to the object you are studying and check them one by one. If a hypothesis fails, either strengthen the object, pass to a quotient or localisation where the theorem applies, or use a more elementary argument.
For computational work, record each transformation together with the equivalence relation it preserves. In algebra, row operations, similarity, quotienting, localisation and isomorphism preserve different kinds of information. A calculation is useful only when the preserved structure matches the question.
Common failure modes
| Failure mode | Control |
|---|---|
| Confusing prime and maximal ideals. | Return to the definition or theorem hypotheses and verify the missing condition before continuing. |
| Assuming localisation preserves every property automatically. | Return to the definition or theorem hypotheses and verify the missing condition before continuing. |
| Using finite-generation conclusions without a chain condition. | Return to the definition or theorem hypotheses and verify the missing condition before continuing. |
| Treating radicals, integral closure or dimension as elementwise notions only. | Return to the definition or theorem hypotheses and verify the missing condition before continuing. |
| Assuming geometric statements over arbitrary fields without checking the field hypotheses. | Return to the definition or theorem hypotheses and verify the missing condition before continuing. |
Verification checklist
- The ambient set, ring, field, group, module or category has been stated.
- Every operation and map used is well-defined in that setting.
- The hypotheses of each structural result have been checked before use.
- Representatives, coordinates or generators have not been confused with the underlying object.
- Existence and uniqueness have been separated where both matter.
- The final result has been checked against the original defining relation or universal property.
Quick questions
What should I identify first in a problem about multivariable division and polynomial ideal reduction bases?
Start with the ambient algebraic structure, its operation or maps, and the exact hypotheses. Most incorrect solutions begin by using a familiar rule that is not valid in the stated structure.
How should definitions be used in proofs?
Expand the definition at the point where it becomes useful. Definitions are not background prose; they are the conditions that determine what must be proved and which implications are available.
When is a structural theorem safer than direct calculation?
Use a structural theorem when its hypotheses are satisfied and the calculation would otherwise depend on arbitrary coordinates, representatives or generators. The theorem usually identifies an invariant that survives those choices.
How can a final answer be checked?
Substitute the result back into the defining relation, verify any required closure or map property, and check edge cases such as zero, the identity, the empty object or degenerate quotients where relevant.