a gentle introduction to elimination theory zafeirakis...

A gentle introduction to Elimination Theory

March 2018 @ METU

Zafeirakis Zafeirakopoulos

Disclaimer

I Elimination theory is a very wide area of research.

I We will see only parts of it

I through the lens of computation (polynomial system solving)

Z.Zafeirakopoulos 2

Disclaimer

Z.Zafeirakopoulos 2

Disclaimer

Z.Zafeirakopoulos 2

Membership

Z.Zafeirakopoulos 3

Membership - A tale of computation

Definition (Ideal)

Given a ring R, an ideal I ∈ R is a subset of R such that

I ∀a ∈ I, ∀c ∈ R : ca ∈ II ∀a, b ∈ I : a + b ∈ I

Definition (Ideal Membership)

Input a ring R, an ideal I ⊆ R and an element f ∈ ROutput True if f ∈ I, False otherwise

Z.Zafeirakopoulos 4

Definition (Ideal)

Given a ring R, an ideal I ∈ R is a subset of R such that

I ∀a ∈ I, ∀c ∈ R : ca ∈ II ∀a, b ∈ I : a + b ∈ I

Definition (Ideal Membership)

Input a ring R, an ideal I ⊆ R and an element f ∈ ROutput True if f ∈ I, False otherwise

Z.Zafeirakopoulos 4

I Membership in Euclidean domains is easy

I Division gives unique remainder

I By division we obtain a linear combination f = r +∑gi∈I

I f ∈ I if and only if r = 0

Note R[x ] is a Euclidean domain

Z.Zafeirakopoulos 4

A(t) =

t 2t 2

t + 1 0 2t

←−

←−−−−−−

−(t+1)

0 t 2− t

0 −t − 1 t − 1

←−+ | · (−1)

0 t 2− t

0 1 −1

←−←− ←−

+ | : 2

1 1 10 1 −10 0 1

def Gauss(M):

for col in range(len(M[0])):

for row in range(col+1, len(M)):

r = [(rowValue * (-(M[row][col] / M[col][col]))) for rowValue in M[col]]

M[row] = [sum(pair) for pair in zip(M[row], r)]

Z.Zafeirakopoulos 4

Emmy Nother1920s

NoteIf we want to manipulate ideals, we have tobe able to decide membership.If we can, then we can also

I decide equality of ideals

I Arithmetic of ideals

Z.Zafeirakopoulos 4

Grete Herman1940s

I Proved that a bound to decide membership would be doublyexponential in the degree.

I The linear combination has huge coefficients.

I Indication that Grobner bases have bad complexity.

Z.Zafeirakopoulos 4

Grete Herman1940s

I Proved that a bound to decide membership would be doublyexponential in the degree.

I The linear combination has huge coefficients.

I Indication that Grobner bases have bad complexity.

Z.Zafeirakopoulos 4

Wolfgang Grobner1940s

I Worked with Nother

I Several contributions

I Did not invent the bases bearing his name

Z.Zafeirakopoulos 4

I Membership is hard because remainder is not unique

I For some sets of divisors, remainder is unique

I Every ideal (in a Notherian ring) has such a set of generators.

I Buchberger proved it.

I What is even better, he proved it constructively.

Z.Zafeirakopoulos 4

def Groebner(ideal):

updated = True

while updated:

updated = False

for f in ideal:

for g in ideal:

r = S_polynomial(f,g).divide(ideal)

if not r.is_zero():

ideal.append(r)

updated = True

if updated: break

return ideal

I Bruno Buchberger was a student of Grobner

Thesis An Algorithm for Finding the Basis Elements of the ResidueClass Ring Modulo a Zerodimensional Polynomial Ideal

Z.Zafeirakopoulos 4

Monomial (Order)

Definition (Term Monoid)

Given a set of variables x1, x2, . . . , xd we consider the multiplicativemonoid T =

{xα11 xα2

2 · · · xαdd : α ∈ Nd

Definition (Term order)

Let ≤ be a total order on T. It is called a term order if

I 0 ≤ T for all T ∈ T and

I if a ≤ b then ac ≤ bc for all a, b, c ∈ T.

Z.Zafeirakopoulos 5

Monomial (Order)

{xα11 xα2

2 · · · xαdd : α ∈ Nd

Note that there is a monoid homomorphism between T and Nd

Z.Zafeirakopoulos 5

Monomial (Order)

{xα11 xα2

2 · · · xαdd : α ∈ Nd

Note that there is a monoid homomorphism between T and Nd

Z.Zafeirakopoulos 5

Monomial (Order)

Example (Lexicographic vs DegRevLex)

Fix x1 ≤ x2 ≤ · · · ≤ xd .

I xα11 xα2

2 · · · xαdd ≤lex xβ11 xβ22 · · · x

βdd if the left-most non-zero

entry in β − α is positive.

I xα11 xα2

2 · · · xαdd ≤drl x

β11 xβ22 · · · x

βdd if

I∑αi ≤

∑βi or

I∑αi =

∑βi and the right-most non-zero entry in β − α is

positive.

Z.Zafeirakopoulos 5

Monomial (Order)

A term order induces an order on (monomials and thus on)polynomials in K[x1, . . . , xd ].

Z.Zafeirakopoulos 5

Grobner Bases

Fix a term order.

DefinitionGiven an ideal I = 〈f1, f2, . . . , fn〉, a Grobner basis for I , withrespect to the term order, is a set G = {g1, g2, . . . , gm} such thatI = 〈G 〉 and for every 0 6= f ∈ I we have that lt (gi ) | lt (f ) forsome i ∈ [m].

I This is not the only definition.

I Other definitions will appear during this series.

Z.Zafeirakopoulos 6

Grobner Bases

Fix a term order.

Z.Zafeirakopoulos 6

Grobner Bases

Fix a term order.

A more important property:

reduction by G in K[x1, . . . , xd ] is unique.

Z.Zafeirakopoulos 6

Grobner Bases

Fix a term order.

A more important property:

reduction by G in K[x1, . . . , xd ] is unique.

Reduce means to take the remainder after we divide as much aspossible with elements of G .Z.Zafeirakopoulos 6

A criterion

Definition (S-polynomial)

Fix a term order and let f , g ∈ K[x1, . . . , xd ]. The S-polynomial off and g is

Sf ,g =lcm (lt(f ), lt(g))

lt(f )f − lcm (lt(f ), lt(g))

lt(g)g

Theorem (Buchberger)

A (finite) set G is a Grobner basis of 〈G 〉 if and only if Sf ,g isreduced to 0 by G for all pairs f , g ∈ G .

Z.Zafeirakopoulos 7

A criterion

Definition (S-polynomial)

Fix a term order and let f , g ∈ K[x1, . . . , xd ]. The S-polynomial off and g is

Sf ,g =lcm (lt(f ), lt(g))

lt(f )f − lcm (lt(f ), lt(g))

lt(g)g

Theorem (Buchberger)

A (finite) set G is a Grobner basis of 〈G 〉 if and only if Sf ,g isreduced to 0 by G for all pairs f , g ∈ G .

Z.Zafeirakopoulos 7

Example

I f1 = x2 + (y − 1)2 − 1

I f2 = y2

and I = 〈f1, f2〉.

G ={x2 − 2y , y2

}Sx2+(y−1)2−1,y2 = y4 − 2y3 = y2(−y2 + 2y)→y2 0

We interreduce elements of the GB to obtain x2 − 2y and y2.We tend to say that a GB is a ”nice” choice of a generators.

Z.Zafeirakopoulos 8

Example

I f1 = x2 + (y − 1)2 − 1

I f2 = y2

and I = 〈f1, f2〉. Then

G ={x2 − 2y , y2

Sx2+(y−1)2−1,y2 = y4 − 2y3 = y2(−y2 + 2y)→y2 0

Z.Zafeirakopoulos 8

Example

I f1 = x2 + (y − 1)2 − 1

I f2 = y2

G ={x2 − 2y , y2

}Sx2+(y−1)2−1,y2 = y4 − 2y3 = y2(−y2 + 2y)→y2 0

Z.Zafeirakopoulos 8

Example

I f1 = x2 + (y − 1)2 − 1

I f2 = y2

G ={x2 − 2y , y2

}Sx2+(y−1)2−1,y2 = y4 − 2y3 = y2(−y2 + 2y)→y2 0

We interreduce elements of the GB to obtain x2 − 2y and y2.

We tend to say that a GB is a ”nice” choice of a generators.

Z.Zafeirakopoulos 8

Example

I f1 = x2 + (y − 1)2 − 1

I f2 = y2

G ={x2 − 2y , y2

}Sx2+(y−1)2−1,y2 = y4 − 2y3 = y2(−y2 + 2y)→y2 0

Z.Zafeirakopoulos 8

Elimination Ideal

Z.Zafeirakopoulos 9

Elimination ideal

DefinitionLet I ⊆ K[x1, . . . , xd ] be an ideal. Then we define the i-thelimination ideal of I as

Ii = I ∩K[xi+1, . . . , xd ].

Theorem (Elimination Property of Grobner Bases)

Let k ∈ [d ] and fix a lexicographic order such that xi ≤ xj for alli < k and k < j . If G is a Grobner basis of I (for the term orderwe fixed), then

Ik = 〈G ∩K[xk+1, . . . , xd ]〉 .

Z.Zafeirakopoulos 10

Elimination ideal

DefinitionLet I ⊆ K[x1, . . . , xd ] be an ideal. Then we define the i-thelimination ideal of I as

Ii = I ∩K[xi+1, . . . , xd ].

Theorem (Elimination Property of Grobner Bases)

Let k ∈ [d ] and fix a lexicographic order such that xi ≤ xj for alli < k and k < j . If G is a Grobner basis of I (for the term orderwe fixed), then

Ik = 〈G ∩K[xk+1, . . . , xd ]〉 .

Elimination ideal

I f1 = x2 + (y − 1)2 − 1

I f2 = y2

and I = 〈f 1, f2〉.

We saw that a GB for I is{x2 − 2y , y2

}. Thus

Ix ={x2 − 2y , y2

}∩K[y ] =

⟨y2⟩

Elimination ideal

I f1 = x2 + (y − 1)2 − 1

I f2 = y2

and I = 〈f 1, f2〉.

We saw that a GB for I is{x2 − 2y , y2

}. Thus

Ix ={x2 − 2y , y2

}∩K[y ] =

⟨y2⟩

Variety and Vanishing Ideal

Let K be an algebraically closed field.

Definition (Variety)

Let I be an ideal in K[x1, . . . , xd ]. Then

V (I ) ={x ∈ Kd : f (x) = 0 for all f ∈ I

Definition (Vanishing)

Let V ⊆ Kd be a variety. Then

I (V ) = {f ∈ K[x1, . . . , xd ] : f (x) = 0 for all x ∈ V }

I What does it mean for the GB that the variety is 0-dim?

I What does it mean for the elimination ideal?

I What does it mean for solving?

I What does it remind us?

Variety of the Elimination Ideal

For f1, . . . , fn ∈ K[x1, . . . , xd ], we write fi in the form

fi = hi (x2, . . . , xd)xNi1 + terms of x1-degree less than Ni ,

for each 1 ≤ i ≤ n. Consider the projection π : Kn → Kn−1:

π((c1, c2, . . . , cn)

)= (c2, c3, . . . , cn) .

Theorem (Elimination Theorem)

Let I1 be the first elimination ideal of an ideal I E K[x1, . . . , xn].Then

V (I1) = π(V (I )

)∪(V (h1, . . . , hm) ∩ V (I1)

Modeling complementary sequences

T = (a, 0, a)|(a, 0,−a)|(0,−a, 0)|(0, b, 0) 0→