pictures of monomial ideals - loras...
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Pictures of Monomial Ideals
Angela Kohlhaas
Bi-State Math Colloquium
February 20, 2013
Why study ideals?
O Solving Equations
O Linear
O Quadratic
O Cubic
O Higher degree?
O Systems of
Equations
O Several variables
O Linear
O Higher degree?
For now, think of ideals as sets of polynomials
Ideals arise in Ring Theory
A ring πΉ (commutative, with identity) is a set
with the following properties:
O Closed under addition and multiplication
O Associative and commutative under
addition and multiplication
O Additive identity (0)
O Additive inverses
O Multiplicative identity (1)
O May NOT have multiplicative inverses
O If all nonzero elements do, itβs called a field.
Examples of Rings
O β, the set of real numbers
O β, the set of rational numbers
O β€, the set of integers
O β€ π₯ , polynomials in one variable with
integer coefficients
O β π₯, π¦ , polynomials in two variables with
real coefficients
Ideals
An ideal π° is a subset of a ring π
satisfying the following property:
O If π, π are in π°, then ππ + ππ is in π° for any π, π in πΉ.
O That is, πΌ is closed under linear combinations with
coefficients in the ring.
O Closed under addition
O Closed under βscalarβ multiplication
Examples of Ideals
O π = β€, πΌ = 5
= 5π βΆ π β β€
O π = β π₯, π¦ , πΌ = π₯2 β π₯π¦, 3π₯ + π¦
= π π₯2 β π₯π¦ + π 3π₯ + π¦ βΆ π, π β π
O π = β π₯, π¦ , πΌ = π₯2, π₯π¦3, π¦5
= ππ₯2 + ππ₯π¦3 + ππ¦5 βΆ π, π, π β π
O Each generator is a monomial, a single term
Rings mimic the Integers
O Prime factorization / Primary decomposition
O In β€, factor 200
O In β π₯, π¦ , factor π₯4π¦ β π₯3π¦2
O What about 200 and π₯2, π₯π¦3, π¦5 ?
O Modular arithmetic / Quotient rings
O β€/(5) = π + 5 βΆ π β β€
O β π₯, π¦ /(π₯2, π₯π¦3, π¦5) = ?
O Allows us to find the dimension or βsizeβ
Pictures!
πΌ = π₯2, π₯π¦3, π¦5
πΌ = π₯2, π₯π¦3, π¦5
Dimension of π /πΌ?
Primary Decomposition
πΌ = x4, x3π¦2, π₯2π¦4, π¦5
πΌ = x4, x3π¦2, π₯2π¦4, π¦5
Primary Decomposition?
What if π = β π₯, π¦, π§ ?
Let πΌ = π₯3, π¦4, π§2, π₯π¦2π§ .
O Diagram: think π /πΌ
O Dimension of π /πΌ?
O Primary Decomposition?
Colon Ideals
Let π» and πΌ be ideals in a ring π
with π» contained in πΌ.
O Then π»: πΌ = π β π βΆ ππΌ β π»
O That is, π»: πΌ is the set of
elements of the ring which
move all elements of πΌ into π».
πΌ = π₯2, π₯π¦3, π¦5 π» = π₯2, π¦5
πΌ = π₯2, π₯π¦3, π¦5 π» = π₯2, π¦5
πΌ = π₯2, π₯π¦3, π¦5 π» = π₯2, π¦5
πΌ = π₯2, π₯π¦3, π¦5 π» = π₯2, π¦5
π»: π½ = π₯, π¦2
πΌ = π₯4, π₯3π¦2, π₯2π¦4, π¦5 π» = π₯4, π¦5
πΌ = π₯4, π₯3π¦2, π₯2π¦4, π¦5 π» = π₯4, π¦5
πΌ = π₯4, π₯3π¦2, π₯2π¦4, π¦5 π» = π₯4, π¦5
Reductions of Ideals
O Reductions are simpler ideals
contained in a larger ideal with
similar properties.
O βSimplerβ usually means fewer
generators.
O Most minimal reductions are not
monomial, but their intersection is.
The Core of an Ideal
O The core of an ideal πΌ is the
intersection of all reductions of πΌ.
O If πΌ is monomial, so is core πΌ .
O Cores have symmetry similar to
colon ideals.
πΌ = π₯2, π₯π¦3, π¦5 core πΌ
πΌ = π₯4, π₯3π¦2, π₯2π¦4, π¦5 core(πΌ)
πΌ = π₯11, π₯9π¦2, π₯6π¦3, π₯5π¦5, π₯4π¦6, π₯2π¦7, π₯π¦9, π¦10
πΌ = π₯11, π₯9π¦2, π₯6π¦3, π₯5π¦5, π₯4π¦6, π₯2π¦7, π₯π¦9, π¦10
πΌ = π₯6, π¦4, π§5, π₯2π¦π§ core πΌ
Why study monomial ideals?
O Can reduce more complicated ideals to
monomial ideals with similar properties
(GrΓΆbner basis theory).
O Monomial ideals can be studied with
combinatorial methods, not just algebraic.
O They are algorithmic, easy to program.