Elimination Theory

Carlos D’Andrea

Fourth EACA International School on Computer Algebra and its Applications

Carlos D’Andrea Elimination Theory What is Elimination Theory?

Carlos D’Andrea Elimination Theory Elimination Theory

Carlos D’Andrea Elimination Theory In the problem of elimination, one seeks the relationship that must exist between the coefficients of a function or system of functions in order that some particular circumstance (or singularity) can occur. Cayley, 1864 Nouvelles recherches sur l’élimination et la théorie des courbes

A bit of history

Carlos D’Andrea Elimination Theory A bit of history

In the problem of elimination, one seeks the relationship that must exist between the coefficients of a function or system of functions in order that some particular circumstance (or singularity) can occur. Cayley, 1864 Nouvelles recherches sur l’élimination et la théorie des courbes

Carlos D’Andrea Elimination Theory A bit of history

A system of arbitrarily many algebraic equations for z0, z0, z00, ..., z(n−1), in which the coefficients belong to the rationality domain( R, R0, R00,....), define the algebraic relations between z and R, whose knowledge and representation are the purpose of the theory of elimination. Kronecker, 1882 Grundzügeeiner arithmetischen Theorie der algebraischen Grössen

Carlos D’Andrea Elimination Theory Find “the condition” on a10, a11, a20, a21 so that the system a10x0 + a11x1 = 0 a20x0 + a21x1 = 0 has a solution diﬀerent from (0, 0)

The baby example in elimination

Carlos D’Andrea Elimination Theory The baby example in elimination

Find “the condition” on a10, a11, a20, a21 so that the system a10x0 + a11x1 = 0 a20x0 + a21x1 = 0 has a solution diﬀerent from (0, 0)

Carlos D’Andrea Elimination Theory ↓ a10a21 − a20a11 ∈ K[a10, a11, a20, a21]

“Elimination”

a10x0 + a11x1, a20x0 + a21x1

∈ K[a10, a11, a20, a21, x0, x1]

Carlos D’Andrea Elimination Theory “Elimination”

a10x0 + a11x1, a20x0 + a21x1

∈ K[a10, a11, a20, a21, x0, x1] ↓ a10a21 − a20a11 ∈ K[a10, a11, a20, a21]

Carlos D’Andrea Elimination Theory to have a solution diﬀerent from (0, 0,..., 0)

More general

Find “the condition” for the system   a10x0 + a11x1 + ... + a1nxn = 0   a20x0 + a21x1 + ... + a2nxn = 0 . . .  . . .   a(n+1)0x0 + a(n+1)1x1 + ... + a(n+1)nxn = 0

Carlos D’Andrea Elimination Theory More general

Find “the condition” for the system   a10x0 + a11x1 + ... + a1nxn = 0   a20x0 + a21x1 + ... + a2nxn = 0 . . .  . . .   a(n+1)0x0 + a(n+1)1x1 + ... + a(n+1)nxn = 0 to have a solution diﬀerent from (0, 0,..., 0)

Carlos D’Andrea Elimination Theory Another more general

Let d1, d2 ∈ N. Find “the condition” for the system of polynomials d d −1 a10x0 1 + a11x0 1 x1 + ... = 0 d d −1 a20x0 2 + a21x0 2 x1 + ... = 0 to have a solution diﬀerent from (0, 0)

Carlos D’Andrea Elimination Theory More more more general...

Let n ∈ N, and d1,..., dn+1 ∈ N, ﬁnd the condition for  P α0 αn a1,α ,...,α x0 ... xn = 0  α0+...+αn=d1 0 n    P α0 αn  a2,α ,...,α x0 ... xn = 0  α0+...+αn=d2 0 n . . .  . . .     P a x α0 ... x αn = 0 α0+...+αn=dn+1 n+1,α0,...,αn 0 n to have a solution diﬀerent from (0, 0,..., 0)

Carlos D’Andrea Elimination Theory Elimination: The general problem

For a = (a1,..., aN ), k, n ∈ N let f1(a, x1,..., xn),..., fk (a, x1,..., xn) ∈ K[a, x1,..., xn]. Find conditions on a such that   f1(a, x1,..., xn) = 0   f2(a, x1,..., xn) = 0 . . .  . . .   fk (a, x1,..., xn) = 0 has a solution

Carlos D’Andrea Elimination Theory There is not necessarily a “closed” condition Algorithms??

Solution?

Depends on the ground ﬁeld

Carlos D’Andrea Elimination Theory Algorithms??

Solution?

Depends on the ground ﬁeld There is not necessarily a “closed” condition

Carlos D’Andrea Elimination Theory Solution?

Depends on the ground ﬁeld There is not necessarily a “closed” condition Algorithms??

Carlos D’Andrea Elimination Theory Conditions: all maximal minors of aij 1≤i≤k, 1≤j≤n equal to zero

Easy example

  a11x1 + ... + a1nxn = 0   a21x1 + ... + a2nxn = 0 . . .  . . .   aknx1 + ... + aknxn = 0 with k ≥ n

Carlos D’Andrea Elimination Theory Easy example

  a11x1 + ... + a1nxn = 0   a21x1 + ... + a2nxn = 0 . . .  . . .   aknx1 + ... + aknxn = 0 with k ≥ n Conditions: all maximal minors of aij 1≤i≤k, 1≤j≤n equal to zero

Carlos D’Andrea Elimination Theory Conditions?

Another “easy” example

k = n = 1, 2 d a0 + a1x1 + a2x1 + ... + ad x1 = 0

Carlos D’Andrea Elimination Theory Another “easy” example

k = n = 1, 2 d a0 + a1x1 + a2x1 + ... + ad x1 = 0 Conditions?

Carlos D’Andrea Elimination Theory V ⊂ KN × Kn

π1|V ↓ ↓ π1

N π1(V ) ⊂ K

The set of conditions is π1(V ) is not necessarily described by zeroes of polynomials

Geometry

V = {(a, x1,..., xn): f1(a, x1,..., xn) = 0,... fk (a, x1,..., xn) = 0}

Carlos D’Andrea Elimination Theory The set of conditions is π1(V ) is not necessarily described by zeroes of polynomials

Geometry

V = {(a, x1,..., xn): f1(a, x1,..., xn) = 0,... fk (a, x1,..., xn) = 0} V ⊂ KN × Kn

π1|V ↓ ↓ π1

N π1(V ) ⊂ K

Carlos D’Andrea Elimination Theory Geometry

V = {(a, x1,..., xn): f1(a, x1,..., xn) = 0,... fk (a, x1,..., xn) = 0} V ⊂ KN × Kn

π1|V ↓ ↓ π1

N π1(V ) ⊂ K

The set of conditions is π1(V ) is not necessarily described by zeroes of polynomials

Carlos D’Andrea Elimination Theory V = {(a, x0, x1,..., xn): f1(a, x0, x1,..., xn) = 0,... fk (a, x0, x1,..., xn) = 0} V ⊂ KN × Pn

π1|V ↓ ↓ π1

N π1(V ) ⊂ K

π1(V ) = {p1(a) = 0,..., p`(a) = 0}

Projective Elimination

(homogeneous polynomials in an algebraically closed ﬁeld)

Carlos D’Andrea Elimination Theory V ⊂ KN × Pn

π1|V ↓ ↓ π1

N π1(V ) ⊂ K

π1(V ) = {p1(a) = 0,..., p`(a) = 0}

Projective Elimination

(homogeneous polynomials in an algebraically closed ﬁeld) V = {(a, x0, x1,..., xn): f1(a, x0, x1,..., xn) = 0,... fk (a, x0, x1,..., xn) = 0}

Carlos D’Andrea Elimination Theory π1(V ) = {p1(a) = 0,..., p`(a) = 0}

Projective Elimination

(homogeneous polynomials in an algebraically closed ﬁeld) V = {(a, x0, x1,..., xn): f1(a, x0, x1,..., xn) = 0,... fk (a, x0, x1,..., xn) = 0} V ⊂ KN × Pn

π1|V ↓ ↓ π1

N π1(V ) ⊂ K

Carlos D’Andrea Elimination Theory Projective Elimination

(homogeneous polynomials in an algebraically closed ﬁeld) V = {(a, x0, x1,..., xn): f1(a, x0, x1,..., xn) = 0,... fk (a, x0, x1,..., xn) = 0} V ⊂ KN × Pn

π1|V ↓ ↓ π1

N π1(V ) ⊂ K

π1(V ) = {p1(a) = 0,..., p`(a) = 0}

Carlos D’Andrea Elimination Theory Input: f1(a, x),... fk(a, x) Output: p1(a),..., p`(a)

How do we compute these conditions?

Carlos D’Andrea Elimination Theory Output: p1(a),..., p`(a)

How do we compute these conditions?

Input: f1(a, x),... fk(a, x)

Carlos D’Andrea Elimination Theory How do we compute these conditions?

Input: f1(a, x),... fk(a, x) Output: p1(a),..., p`(a)

Carlos D’Andrea Elimination Theory Set s := max deg(fi ), and G0,..., Gn expressions of the form P α Gj (a, x) := uα,i x fi (a, x) of degree s in x with new parameters uα,i . Then, “the Kronecker u-Resultant” P β Resx (G0,..., Gn) = β Pβ(a)u gives the conditions {Pβ(a)}

Mertens 1889

Carlos D’Andrea Elimination Theory Then, “the Kronecker u-Resultant” P β Resx (G0,..., Gn) = β Pβ(a)u gives the conditions {Pβ(a)}

Mertens 1889

Set s := max deg(fi ), and G0,..., Gn expressions of the form P α Gj (a, x) := uα,i x fi (a, x) of degree s in x with new parameters uα,i .

Carlos D’Andrea Elimination Theory Mertens 1889

Set s := max deg(fi ), and G0,..., Gn expressions of the form P α Gj (a, x) := uα,i x fi (a, x) of degree s in x with new parameters uα,i . Then, “the Kronecker u-Resultant” P β Resx (G0,..., Gn) = β Pβ(a)u gives the conditions {Pβ(a)} Carlos D’Andrea Elimination Theory For arbitrary s, we use the “Macaulay matrix” Ms(a) such that  .   .  . . β α Ms(a)  x  =  x fi  , |α| = s  .   .  . .

For s 0, all maximal minors of Ms(a) vanish iﬀ there is a common solution

van der Waerden 1926

Carlos D’Andrea Elimination Theory For s 0, all maximal minors of Ms(a) vanish iﬀ there is a common solution

van der Waerden 1926

For arbitrary s, we use the “Macaulay matrix” Ms(a) such that  .   .  . . β α Ms(a)  x  =  x fi  , |α| = s  .   .  . .

Carlos D’Andrea Elimination Theory van der Waerden 1926

For arbitrary s, we use the “Macaulay matrix” Ms(a) such that  .   .  . . β α Ms(a)  x  =  x fi  , |α| = s  .   .  . .

For s 0, all maximal minors of Ms(a) vanish iﬀ there is a common solution Carlos D’Andrea Elimination Theory I π1(V ) = ∞ (hf1,..., fki : hx0,..., xni ) ∩ K[a] The “elimination ideal”

Algebra & Geometry

(van der Waerden, 1926)

Carlos D’Andrea Elimination Theory ∩ K[a] The “elimination ideal”

Algebra & Geometry

(van der Waerden, 1926)

I π1(V ) = ∞ (hf1,..., fki : hx0,..., xni )

Carlos D’Andrea Elimination Theory Algebra & Geometry

(van der Waerden, 1926)

I π1(V ) = ∞ (hf1,..., fki : hx0,..., xni ) ∩ K[a] The “elimination ideal”

Carlos D’Andrea Elimination Theory the polynomials are not homogeneous? K is not algebraically closed?

But what if...

Carlos D’Andrea Elimination Theory K is not algebraically closed?

But what if...

the polynomials are not homogeneous?

Carlos D’Andrea Elimination Theory But what if...

the polynomials are not homogeneous? K is not algebraically closed?

Carlos D’Andrea Elimination Theory Embed K ,→ K and work there Use speciﬁc (topological, characteristic,...) properties of K ...

K is the problem?

Carlos D’Andrea Elimination Theory Use speciﬁc (topological, characteristic,...) properties of K ...

K is the problem?

Embed K ,→ K and work there

Carlos D’Andrea Elimination Theory ...

K is the problem?

Embed K ,→ K and work there Use speciﬁc (topological, characteristic,...) properties of K

Carlos D’Andrea Elimination Theory K is the problem?

Embed K ,→ K and work there Use speciﬁc (topological, characteristic,...) properties of K ...

Carlos D’Andrea Elimination Theory Analogy with Linear Algebra...

Homogeneous vs non homogeneous

Carlos D’Andrea Elimination Theory Homogeneous vs non homogeneous

Analogy with Linear Algebra...

Carlos D’Andrea Elimination Theory V ⊂ KN × Pn

π1|V ↓ ↓ π1

N π1(V ) ⊂ K

π1(V ) = {p1(a) = 0} One hopes!

Generalization of the determinant?

V = {(a, x0, x1,..., xn): f1(a, x0, x1,..., xn) = 0,..., fn+1(a, x0, x1,..., xn) = 0}

Carlos D’Andrea Elimination Theory π1(V ) = {p1(a) = 0} One hopes!

Generalization of the determinant?

V = {(a, x0, x1,..., xn): f1(a, x0, x1,..., xn) = 0,..., fn+1(a, x0, x1,..., xn) = 0} V ⊂ KN × Pn

π1|V ↓ ↓ π1

N π1(V ) ⊂ K

Carlos D’Andrea Elimination Theory One hopes!

Generalization of the determinant?

V = {(a, x0, x1,..., xn): f1(a, x0, x1,..., xn) = 0,..., fn+1(a, x0, x1,..., xn) = 0} V ⊂ KN × Pn

π1|V ↓ ↓ π1

N π1(V ) ⊂ K

π1(V ) = {p1(a) = 0}

Carlos D’Andrea Elimination Theory Generalization of the determinant?

V = {(a, x0, x1,..., xn): f1(a, x0, x1,..., xn) = 0,..., fn+1(a, x0, x1,..., xn) = 0} V ⊂ KN × Pn

π1|V ↓ ↓ π1

N π1(V ) ⊂ K

π1(V ) = {p1(a) = 0} One hopes!

Carlos D’Andrea Elimination Theory p1(a) = det(aij)

Example 1

 a00x0 + a01x1 + ... + a0nxn = 0   a10x0 + a11x1 + ... + a1nxn = 0 . . .   an0x0 + an1x1 + ... + annxn = 0

Carlos D’Andrea Elimination Theory Example 1

 a00x0 + a01x1 + ... + a0nxn = 0   a10x0 + a11x1 + ... + a1nxn = 0 . . .   an0x0 + an1x1 + ... + annxn = 0

p1(a) = det(aij)

Carlos D’Andrea Elimination Theory   a10 a11 ... a1d1 0 ... 0

 0 a10 ... a1d1−1 a1d1 ... 0   . . .   ......   ......   0 0 ... a ...... a  Res(f , f ) = det  10 1d1  1 2  a a ... a 0 ... 0   20 21 2d2   0 a ... a a ... 0   20 2d2−1 2d2   . . . . .   ...... 

0 0 ... a20 ...... a2d2

Example 2

d1 d1−1 d1 f1 = a10x0 + a11x0 x1 + ... + a1d1 x1 d2 d2−1 d2 f2 = a20x0 + a21x0 x1 + ... + a2d2 x1

Carlos D’Andrea Elimination Theory Example 2

d1 d1−1 d1 f1 = a10x0 + a11x0 x1 + ... + a1d1 x1 d2 d2−1 d2 f2 = a20x0 + a21x0 x1 + ... + a2d2 x1   a10 a11 ... a1d1 0 ... 0

0 0 ... a20 ...... a2d2 Carlos D’Andrea Elimination Theory Resd1,...,dn (f1, f2,..., fn+1) Macaulay/dense/classical resultant

Example 3

 P α0 αn f1 = a1,α ,...,α x0 ... xn  α0+...+αn=d1 0 n  P α0 αn  f2 = α +...+α =d a2,α0,...,αn x0 ... xn . 0 n 2 .   f = P a x α0 ... x αn n+1 α0+...+αn=dn+1 n+1,α0,...,αn 0 n

Carlos D’Andrea Elimination Theory Macaulay/dense/classical resultant

Example 3

Resd1,...,dn (f1, f2,..., fn+1)

Carlos D’Andrea Elimination Theory Example 3

Resd1,...,dn (f1, f2,..., fn+1) Macaulay/dense/classical resultant

Carlos D’Andrea Elimination Theory  4 2 2 4 a00x0 + a01x0 x1x2 + a02x1 x2 = 0  4 2 2 4 a10x0 + a11x0 x1x2 + a12x1 x2 = 0  4 2 2 4 a20x0 + a21x0 x1x2 + a22x1 x2 = 0

p1(a) 6= det(aij ) 2 2 2 p1(a) = a00a12 − a00a01a11a12 − 2a00a02a10a12 + a00a02a11 + 2 2 2 a01a10a12 − a01a02a10a11 + a02a10 2 2 2 p2(a) = a00a22 − a00a01a21a22 − 2a00a02a20a22 + a00a02a21 + 2 2 2 a01a20a22 − a01a02a20a21 + a02a20 2 2 2 p3(a) = a20a12 − a20a21a11a12 − 2a20a22a10a12 + a20a22a11 + 2 2 2 a21a10a12 − a21a22a10a11 + a22a10

No-example

Carlos D’Andrea Elimination Theory p1(a) 6= det(aij ) 2 2 2 p1(a) = a00a12 − a00a01a11a12 − 2a00a02a10a12 + a00a02a11 + 2 2 2 a01a10a12 − a01a02a10a11 + a02a10 2 2 2 p2(a) = a00a22 − a00a01a21a22 − 2a00a02a20a22 + a00a02a21 + 2 2 2 a01a20a22 − a01a02a20a21 + a02a20 2 2 2 p3(a) = a20a12 − a20a21a11a12 − 2a20a22a10a12 + a20a22a11 + 2 2 2 a21a10a12 − a21a22a10a11 + a22a10

No-example

 4 2 2 4 a00x0 + a01x0 x1x2 + a02x1 x2 = 0  4 2 2 4 a10x0 + a11x0 x1x2 + a12x1 x2 = 0  4 2 2 4 a20x0 + a21x0 x1x2 + a22x1 x2 = 0

Carlos D’Andrea Elimination Theory 2 2 2 p1(a) = a00a12 − a00a01a11a12 − 2a00a02a10a12 + a00a02a11 + 2 2 2 a01a10a12 − a01a02a10a11 + a02a10 2 2 2 p2(a) = a00a22 − a00a01a21a22 − 2a00a02a20a22 + a00a02a21 + 2 2 2 a01a20a22 − a01a02a20a21 + a02a20 2 2 2 p3(a) = a20a12 − a20a21a11a12 − 2a20a22a10a12 + a20a22a11 + 2 2 2 a21a10a12 − a21a22a10a11 + a22a10

No-example

 4 2 2 4 a00x0 + a01x0 x1x2 + a02x1 x2 = 0  4 2 2 4 a10x0 + a11x0 x1x2 + a12x1 x2 = 0  4 2 2 4 a20x0 + a21x0 x1x2 + a22x1 x2 = 0

p1(a) 6= det(aij )

Carlos D’Andrea Elimination Theory 2 2 2 p2(a) = a00a22 − a00a01a21a22 − 2a00a02a20a22 + a00a02a21 + 2 2 2 a01a20a22 − a01a02a20a21 + a02a20 2 2 2 p3(a) = a20a12 − a20a21a11a12 − 2a20a22a10a12 + a20a22a11 + 2 2 2 a21a10a12 − a21a22a10a11 + a22a10

No-example

 4 2 2 4 a00x0 + a01x0 x1x2 + a02x1 x2 = 0  4 2 2 4 a10x0 + a11x0 x1x2 + a12x1 x2 = 0  4 2 2 4 a20x0 + a21x0 x1x2 + a22x1 x2 = 0

p1(a) 6= det(aij ) 2 2 2 p1(a) = a00a12 − a00a01a11a12 − 2a00a02a10a12 + a00a02a11 + 2 2 2 a01a10a12 − a01a02a10a11 + a02a10

Carlos D’Andrea Elimination Theory 2 2 2 p3(a) = a20a12 − a20a21a11a12 − 2a20a22a10a12 + a20a22a11 + 2 2 2 a21a10a12 − a21a22a10a11 + a22a10

No-example

 4 2 2 4 a00x0 + a01x0 x1x2 + a02x1 x2 = 0  4 2 2 4 a10x0 + a11x0 x1x2 + a12x1 x2 = 0  4 2 2 4 a20x0 + a21x0 x1x2 + a22x1 x2 = 0

Carlos D’Andrea Elimination Theory No-example

 4 2 2 4 a00x0 + a01x0 x1x2 + a02x1 x2 = 0  4 2 2 4 a10x0 + a11x0 x1x2 + a12x1 x2 = 0  4 2 2 4 a20x0 + a21x0 x1x2 + a22x1 x2 = 0

p1(a) =6 det(aij) 2 2 p1(a) = a00a12 − a00a01a11a12 − 2a00a02a10a12 + 2 2 2 2 a00a02a11 + a01a10a12 − a01a02a10a11 + a02a10

Cautionary Example

Carlos D’Andrea Elimination Theory p1(a) =6 det(aij) 2 2 p1(a) = a00a12 − a00a01a11a12 − 2a00a02a10a12 + 2 2 2 2 a00a02a11 + a01a10a12 − a01a02a10a11 + a02a10

Cautionary Example

 2 2 a00x0 + a01x0x1 + a02x1 = 0  2 2 a10x0 + a11x0x1 + a12x1 = 0  a20x0 + a21x1 + a22x2 = 0

Carlos D’Andrea Elimination Theory 2 2 p1(a) = a00a12 − a00a01a11a12 − 2a00a02a10a12 + 2 2 2 2 a00a02a11 + a01a10a12 − a01a02a10a11 + a02a10

Cautionary Example

 2 2 a00x0 + a01x0x1 + a02x1 = 0  2 2 a10x0 + a11x0x1 + a12x1 = 0  a20x0 + a21x1 + a22x2 = 0

p1(a) =6 det(aij)

Carlos D’Andrea Elimination Theory Cautionary Example

 2 2 a00x0 + a01x0x1 + a02x1 = 0  2 2 a10x0 + a11x0x1 + a12x1 = 0  a20x0 + a21x1 + a22x2 = 0

p1(a) =6 det(aij) 2 2 p1(a) = a00a12 − a00a01a11a12 − 2a00a02a10a12 + 2 2 2 2 a00a02a11 + a01a10a12 − a01a02a10a11 + a02a10

Carlos D’Andrea Elimination Theory Two ways: Gr¨obnerBases (w.r.t. “lex”) Triangular Decomposition Both use some kind of division of polynomials

“Triangulating” a system

Carlos D’Andrea Elimination Theory Gr¨obnerBases (w.r.t. “lex”) Triangular Decomposition Both use some kind of division of polynomials

“Triangulating” a system Two ways:

Carlos D’Andrea Elimination Theory Triangular Decomposition Both use some kind of division of polynomials

“Triangulating” a system Two ways: Gr¨obnerBases (w.r.t. “lex”)

Carlos D’Andrea Elimination Theory Both use some kind of division of polynomials

“Triangulating” a system Two ways: Gr¨obnerBases (w.r.t. “lex”) Triangular Decomposition

Carlos D’Andrea Elimination Theory “Triangulating” a system Two ways: Gr¨obnerBases (w.r.t. “lex”) Triangular Decomposition Both use some kind of division of polynomials

Carlos D’Andrea Elimination Theory Every ideal in K[a, x]has a finite set of generators I = hg1,..., gNi {g1,..., gN} is a Gröbnerbasis iff

g ∈ I ⇐⇒ rg1,...gN (g) = 0

Gr¨obnerBases

Carlos D’Andrea Elimination Theory {g1,..., gN} is a Gr¨obnerbasis iﬀ

g ∈ I ⇐⇒ rg1,...gN (g) = 0

Gr¨obnerBases

Every ideal in K[a, x]has a ﬁnite set of generators I = hg1,..., gNi

Carlos D’Andrea Elimination Theory Gr¨obnerBases

Every ideal in K[a, x]has a finite set of generators I = hg1,..., gNi {g1,..., gN} is a Gröbnerbasis iff

g ∈ I ⇐⇒ rg1,...gN (g) = 0

Carlos D’Andrea Elimination Theory In K[a, x], if a ≺ x, then G = GB(I ) =⇒ G ∩ K[a] = GB(I ∩ K[a]) And I ∩ K[a] is the (aﬃne) ideal of the projection

Gr¨obnerBases and Elimination

Carlos D’Andrea Elimination Theory G = GB(I ) =⇒ G ∩ K[a] = GB(I ∩ K[a]) And I ∩ K[a] is the (aﬃne) ideal of the projection

Gr¨obnerBases and Elimination

In K[a, x], if a ≺ x, then

Carlos D’Andrea Elimination Theory And I ∩ K[a] is the (aﬃne) ideal of the projection

Gr¨obnerBases and Elimination

In K[a, x], if a ≺ x, then G = GB(I ) =⇒ G ∩ K[a] = GB(I ∩ K[a])

Carlos D’Andrea Elimination Theory Gr¨obnerBases and Elimination

In K[a, x], if a ≺ x, then G = GB(I ) =⇒ G ∩ K[a] = GB(I ∩ K[a]) And I ∩ K[a] is the (aﬃne) ideal of the projection

Carlos D’Andrea Elimination Theory Characteristic set’s method (J.F. Ritt - 1940’s) Wu’s method (Wenjun Wu - 1970’s)

Triangular sets

Carlos D’Andrea Elimination Theory Wu’s method (Wenjun Wu - 1970’s)

Triangular sets

Characteristic set’s method (J.F. Ritt - 1940’s)

Carlos D’Andrea Elimination Theory Triangular sets

Characteristic set’s method (J.F. Ritt - 1940’s) Wu’s method (Wenjun Wu - 1970’s)

Carlos D’Andrea Elimination Theory For any ideal I ⊂ K[a, x] one can associate a ﬁnite family of characteristic sets of polynomials which are in triangular shape: polynomials in this set have diﬀerent main variables Membership to I is easily tested

Triangular sets

Carlos D’Andrea Elimination Theory polynomials in this set have diﬀerent main variables Membership to I is easily tested

Triangular sets

For any ideal I ⊂ K[a, x] one can associate a ﬁnite family of characteristic sets of polynomials which are in triangular shape:

Carlos D’Andrea Elimination Theory Membership to I is easily tested

Triangular sets

For any ideal I ⊂ K[a, x] one can associate a ﬁnite family of characteristic sets of polynomials which are in triangular shape: polynomials in this set have diﬀerent main variables

Carlos D’Andrea Elimination Theory Triangular sets

For any ideal I ⊂ K[a, x] one can associate a ﬁnite family of characteristic sets of polynomials which are in triangular shape: polynomials in this set have diﬀerent main variables Membership to I is easily tested

Carlos D’Andrea Elimination Theory Gauss elimination Gr¨obnerBases Triangulation Triangular sets Determinants Resultants Cramer’s rule u-resultants ......

The “elimination” dictionary

Linear Polynomial

Carlos D’Andrea Elimination Theory Triangulation Triangular sets Determinants Resultants Cramer’s rule u-resultants ......

The “elimination” dictionary

Linear Polynomial Gauss elimination Gr¨obnerBases

Carlos D’Andrea Elimination Theory Determinants Resultants Cramer’s rule u-resultants ......

The “elimination” dictionary

Linear Polynomial Gauss elimination Gr¨obnerBases Triangulation Triangular sets

Carlos D’Andrea Elimination Theory Cramer’s rule u-resultants ......

The “elimination” dictionary

Linear Polynomial Gauss elimination Gr¨obnerBases Triangulation Triangular sets Determinants Resultants

Carlos D’Andrea Elimination Theory The “elimination” dictionary

Linear Polynomial Gauss elimination Gr¨obnerBases Triangulation Triangular sets Determinants Resultants Cramer’s rule u-resultants ......

Carlos D’Andrea Elimination Theory History Continues...

The rigorous foundation of the theory of algebraic varieties . . . can be formulated more simply than it has been done so far, without the help of elimination theory, on the sole basis of ﬁeld theory and of the general theory of ideals in ring domains. van der Waerden, 1926 First Paper on Algebraic Geometry

Carlos D’Andrea Elimination Theory Work continued in the Bourbaki era: Bruno Buchberger, 1965 Abhyankar, 1976 Giraud 1977 Lazard 1977 Wu 1978 Jouanolou 1979

Fortunately....

Carlos D’Andrea Elimination Theory Bruno Buchberger, 1965 Abhyankar, 1976 Giraud 1977 Lazard 1977 Wu 1978 Jouanolou 1979

Fortunately....

Work continued in the Bourbaki era:

Carlos D’Andrea Elimination Theory Abhyankar, 1976 Giraud 1977 Lazard 1977 Wu 1978 Jouanolou 1979

Fortunately....

Work continued in the Bourbaki era: Bruno Buchberger, 1965

Carlos D’Andrea Elimination Theory Giraud 1977 Lazard 1977 Wu 1978 Jouanolou 1979

Fortunately....

Work continued in the Bourbaki era: Bruno Buchberger, 1965 Abhyankar, 1976

Carlos D’Andrea Elimination Theory Lazard 1977 Wu 1978 Jouanolou 1979

Fortunately....

Work continued in the Bourbaki era: Bruno Buchberger, 1965 Abhyankar, 1976 Giraud 1977

Carlos D’Andrea Elimination Theory Wu 1978 Jouanolou 1979

Fortunately....

Work continued in the Bourbaki era: Bruno Buchberger, 1965 Abhyankar, 1976 Giraud 1977 Lazard 1977

Carlos D’Andrea Elimination Theory Jouanolou 1979

Fortunately....

Work continued in the Bourbaki era: Bruno Buchberger, 1965 Abhyankar, 1976 Giraud 1977 Lazard 1977 Wu 1978

Carlos D’Andrea Elimination Theory Fortunately....

Work continued in the Bourbaki era: Bruno Buchberger, 1965 Abhyankar, 1976 Giraud 1977 Lazard 1977 Wu 1978 Jouanolou 1979

Carlos D’Andrea Elimination Theory Complexity gets into the picture (Pan, Roy, Heintz,...) Computer Aided Design (Sederberg & Goldman) ISSAC conferences start...

In the 80’s

Carlos D’Andrea Elimination Theory Computer Aided Design (Sederberg & Goldman) ISSAC conferences start...

In the 80’s

Complexity gets into the picture (Pan, Roy, Heintz,...)

Carlos D’Andrea Elimination Theory ISSAC conferences start...

In the 80’s

Complexity gets into the picture (Pan, Roy, Heintz,...) Computer Aided Design (Sederberg & Goldman)

Carlos D’Andrea Elimination Theory In the 80’s

Complexity gets into the picture (Pan, Roy, Heintz,...) Computer Aided Design (Sederberg & Goldman) ISSAC conferences start...

Carlos D’Andrea Elimination Theory Algebraic Geometry and Commutative Algebra get involved (Galligo, Eisenbud, Sturmfels, Cox, Gelfand...) Complexity over the reals (Shub & Smale,...) Symbolic-Numeric Methods (Wampler & Sommesse,...) Software to do computation: Macaulay2, CoCoA, Singular,.... MEGA, EACA, ACA, FoCM...

The boom of the 90’s

Carlos D’Andrea Elimination Theory Complexity over the reals (Shub & Smale,...) Symbolic-Numeric Methods (Wampler & Sommesse,...) Software to do computation: Macaulay2, CoCoA, Singular,.... MEGA, EACA, ACA, FoCM...

The boom of the 90’s

Algebraic Geometry and Commutative Algebra get involved (Galligo, Eisenbud, Sturmfels, Cox, Gelfand...)

Carlos D’Andrea Elimination Theory Symbolic-Numeric Methods (Wampler & Sommesse,...) Software to do computation: Macaulay2, CoCoA, Singular,.... MEGA, EACA, ACA, FoCM...

The boom of the 90’s

Algebraic Geometry and Commutative Algebra get involved (Galligo, Eisenbud, Sturmfels, Cox, Gelfand...) Complexity over the reals (Shub & Smale,...)

Carlos D’Andrea Elimination Theory Software to do computation: Macaulay2, CoCoA, Singular,.... MEGA, EACA, ACA, FoCM...

The boom of the 90’s

Algebraic Geometry and Commutative Algebra get involved (Galligo, Eisenbud, Sturmfels, Cox, Gelfand...) Complexity over the reals (Shub & Smale,...) Symbolic-Numeric Methods (Wampler & Sommesse,...)

Carlos D’Andrea Elimination Theory MEGA, EACA, ACA, FoCM...

The boom of the 90’s

Algebraic Geometry and Commutative Algebra get involved (Galligo, Eisenbud, Sturmfels, Cox, Gelfand...) Complexity over the reals (Shub & Smale,...) Symbolic-Numeric Methods (Wampler & Sommesse,...) Software to do computation: Macaulay2, CoCoA, Singular,....

Carlos D’Andrea Elimination Theory The boom of the 90’s

Carlos D’Andrea Elimination Theory Applications dominate the picture SIAM AG, ICIAM... More “general” resultants (Bus´e-D-Sombra,...) Rees Algebras (Cox, Kustin, Polini, Ulrich,...) Representation Matrices (Bus´e,Chardin, Dickenstein,...) ...