Algebraic Geometry - Part-II¶
Table of Contents¶
1. Ideals and Ideal Membership¶
Definition (Ideal): A subset \(I \subseteq \mathbb{K}[x_1,\dots,x_n]\) is an ideal if
- \(0 \in I\) and \(f, g \in I \Rightarrow f + g \in I\) (closed under addition)
- \(f \in I,\ h \in \mathbb{K}[x] \Rightarrow h \cdot f \in I\) (absorbs multiplication)
The ideal generated by \(f_1, \dots, f_s\) is
Hilbert's Basis Theorem guarantees every ideal of \(\mathbb{K}[x_1,\dots,x_n]\) has a finite generating set.
from src.alggeom.polynomial import Variable, Polynomial, parse_polynomial_string
from src.alggeom.ideal import Ideal
x, y = Variable("x"), Variable("y")
v = lambda name: Polynomial.variable(Variable(name))
one = Polynomial.constant
f1 = v('x') * v('y') - one(1) # xy - 1
f2 = v('x')**2 - v('y') # x^2 - y
I = Ideal([f1, f2])
print(I) # Ideal(2 generators)
The central computational question is ideal membership: given \(f\), decide whether \(f \in I\). Since \(f \in I\) iff the remainder of \(f\) modulo a Groebner basis of \(I\) is zero (see Section 5), contains answers it exactly:
print(I.contains(v('y')**2 - v('x'))) # True <- reduces to 0 modulo the basis
print(I.contains(v('x') - v('y'))) # False <- nonzero remainder
2. Ideal Arithmetic: Sum, Product, Intersection¶
Given ideals \(I = \langle F \rangle\) and \(J = \langle G \rangle\):
| Operation | Definition | Generators |
|---|---|---|
| Sum\(I + J\) | \(\{\, f + g : f \in I, g \in J \,\}\) | \(F \cup G\) |
| Product\(IJ\) | \(\langle\, fg : f \in I, g \in J \,\rangle\) | \(\{\, fg : f \in F, g \in G \,\}\) |
| Intersection\(I \cap J\) | \(\{\, f : f \in I \text{ and } f \in J \,\}\) | needs elimination theory |
Ix, Iy = Ideal([v('x')]), Ideal([v('y')])
print([str(g) for g in Ix.sum(Iy).generators]) # ['x', 'y']
print([str(g) for g in Ix.product(Iy).generators]) # ['x * y']
from src.alggeom.groebnerbasis import ideal_intersection
print([str(g) for g in ideal_intersection([v('x')], [v('y')], ['x', 'y'])])
# ['x * y'] <- <x> n <y> = <xy>
Intersection is computed by the classic trick: introduce a new variable \(t\) and eliminate it from
The polynomials of a lex Groebner basis that do not contain \(t\) generate exactly \(I \cap J\).
3. Leading Term Ideals¶
Fix a monomial order. For an ideal \(I\):
This is a monomial ideal - generated by monomials instead of arbitrary polynomials - and its combinatorics encode the geometry of \(V(I)\) (dimension, degree, Hilbert functions).
The generators of \(I\) alone usually do not generate \(\langle \text{LT}(I)\rangle\): reducing one polynomial can produce new leading terms. This is what makes Groebner bases necessary.
4. S-Polynomials¶
Definition (S-polynomial): For nonzero \(f, g\) with leading monomials \(\text{LM}(f)\), \(\text{LM}(g)\), let \(L = \text{lcm}(\text{LM}(f), \text{LM}(g))\). Then
\(S(f,g)\) is constructed so its leading terms cancel. It measures whether the leading terms of \(f\) and \(g\) "interact": if some combination of \(f\) and \(g\) hides a leading term not divisible by either \(\text{LM}(f)\) or \(\text{LM}(g)\), the S-polynomial exposes it.
5. Buchberger's Algorithm and Groebner Bases¶
Definition (Groebner basis): A finite set \(G \subseteq I\) is a Groebner basis of \(I\) if the leading terms of \(G\) generate the leading term ideal:
Equivalently: every \(f \in I\) reduces to \(0\) modulo \(G\).
Buchberger's algorithm starts from the generators and repeatedly:
- Picks a pair \((g_i, g_j)\)
- Computes \(S(g_i, g_j)\)
- Reduces it modulo the current basis
- If the remainder is nonzero, appends it as a new generator
until all S-polynomials reduce to zero. Two classical criteria avoid useless pairs:
- Product criterion - if \(\text{LM}(g_i)\) and \(\text{LM}(g_j)\) share no variable, \(S(g_i,g_j)\) always reduces to zero
- Chain criterion - if some third \(\text{LM}(g_k)\) divides \(\text{lcm}(\text{LM}(g_i), \text{LM}(g_j))\), the pair is redundant
from src.alggeom.groebnerbasis import GroebnerBasis, compute_groebner_basis
gb = GroebnerBasis([f1, f2]) # grevlex by default
print([str(g) for g in gb.groebner_basis])
# ['x * y - 1.0000', 'x^2 - y', 'y^2 - x']
print(gb.is_groebner_basis()) # True (all S-polys reduce to 0)
mem, remainder = gb.ideal_membership(v('y')**2 - v('x'))
print(mem) # True
A basis is called reduced when no generator's leading monomial is divisible by another's and every generator is fully reduced by the others - this form is unique for a given ideal and order.
6. Elimination Theory¶
Elimination Theorem: Let \(G\) be a Groebner basis of \(I \subseteq \mathbb{K}[x_1,\dots,x_n]\) with respect to lex order \(x_1 > x_2 > \dots > x_n\). Then
is a Groebner basis of the \(k\)-th elimination ideal \(I_k = I \cap \mathbb{K}[x_{k+1},\dots,x_n]\).
# I = <x - 1, x + y - 2>; eliminate x
G_lex = compute_groebner_basis(
[v('x') - one(1), v('x') + v('y') - one(2)],
'lex'
)
print([str(g) for g in G_lex])
# ['x - 1.0000', '-y + 1.0000'] <- second poly involves only y
Elimination converts solving systems into finding roots of univariate polynomials - the engine behind solve_polynomial_system in Part III.
7. Saturation and Radical Membership¶
Definition (Saturation):
Geometrically, saturation removes from \(V(I)\) the components lying inside \(\{f = 0\}\).
Definition (Radical):
Membership in \(\sqrt{I}\) is decided by the Rabinowitsch trick: \(f \in \sqrt{I}\) if and only if the ideal \(I + \langle 1 - t f \rangle \subseteq \mathbb{K}[x, t]\) contains \(1\).
from src.alggeom.groebnerbasis import saturation, radical_membership
sat = saturation([v('x')**2 * v('y')], v('x'))
print([str(g) for g in sat])
# ['-y'] <- sat(<x^2*y>, x) = <y>: the x-factor was saturated away
print(radical_membership(v('x'), [v('x')**2]))
# True <- x^2 vanishes on {x=0}, so does x
print(radical_membership(v('x') + one(1), [v('x')**2]))
# False <- x+1 does not vanish on V(x^2)
Ideal.quotient(h) computes \(I : h\) via saturation, and Ideal.radical exposes the radical interface.
8. Primary Decomposition and Primality¶
Definition (Primary decomposition): Every ideal \(I\) can be written as a finite intersection
of primary ideals. Geometrically this decomposes \(V(I)\) into its irreducible components.
The implementation splits using factorization heuristics (common variable factors such as \(xy = x \cdot y\)) combined with saturation, then removes redundant components.
from src.alggeom.groebnerbasis import primary_decomposition
decomp = primary_decomposition([v('x') * v('y')])
print([[str(g) for g in comp] for comp in decomp])
# [['x'], ['y']] <- V(xy) = V(x) u V(y)
print(Ideal([v('x')]).is_prime()) # True <- V(x) irreducible
print(Ideal([v('x') * v('y')]).is_prime()) # False <- two components
An ideal is prime iff its decomposition has a single component (\(I\) prime \(\Leftrightarrow\) \(\mathbb{K}[x]/I\) is an integral domain \(\Leftrightarrow\) \(V(I)\) irreducible).
9. Dimension¶
Krull dimension of the quotient ring, \(\dim(\mathbb{K}[x]/I)\), equals the maximal number of algebraically independent coordinates on \(V(I)\). Computationally it is read off from the initial ideal: the dimension is the size of the largest set \(S\) of variables such that no leading monomial of the Groebner basis uses only variables from \(S\).
d0 = Ideal([v('x') - one(1), v('y') - one(2)]) # two points
dc = Ideal([v('z') - v('x')**2, v('y') - v('x')**2]) # space curve
print(d0.dimension()) # 0 <- finite solution set
print(dc.dimension()) # 1 <- parametrized by x = t
10. The Macaulay Matrix¶
Buchberger processes pairs one at a time. Modern F4-style algorithms instead reduce many S-polynomials simultaneously as linear algebra: stack the polynomials into a Macaulay matrix
with columns ordered by the monomial order, then row-reduce.
import numpy as np
from src.alggeom.groebnerbasis import build_macaulay_matrix
M, monomials = build_macaulay_matrix([f1, f2])
print([str(m) for m in monomials])
# ['x^2', 'x * y', 'y', '1']
print(M.real)
# [[ 0. 1. 0. -1.] <- xy - 1
# [ 1. 0. -1. 0.]] <- x^2 - y
Row reduction of this matrix performs many Gaussian eliminations at once - the foundation for scaling Groebner bases to larger systems.