bases de gröbner e aplicações em otimização

8/20/2019 Bases de Gröbner e Aplicações em Otimização

1/48


2/48


3/48


4/48


5/48

(M, +)

M

+

M

(M, +)

a,b,c

M

a + (b + c) = (a + b) + c

id+ ∈ M id+ + m = m

m + id+ = m m M

(M, +)

u

M

y

M

y + u = id+ u + y = id+ U (M )

M

U (M ) = M

(M, +)

(M, +)

a + b = b + a

a, b

M

(M, +)

+

x ∈ M

−x

(M, +, ·)

+

·

M

• (M, +)

• (M, ·)

•

a,b,c ∈ M a · (b + c) = a · b + a · c

(a + b) · c = a · c + b · c

(M, ·)

(M, ·)

a · b = id+ ⇒ a = id+ b = id+ (M \ {0}, ·)


6/48

+

·

0

1

k ∈ N

Nk := {1, 2, . . . , k}

M

S ⊂ M

S

S = ∅

a, b ∈ M

a − b ∈ M

ab ∈ M

A

n

K

(K[x1, . . . , xn], +, ·) n

K

n

A

n

A

A[x1, . . . , xn]

A[n]

ri=0 aix

αi

r ∈ N αi =

(αi1, . . . , αin) xαi =

n j=1 x j

αij

αi j ∈ N ∪ {0}

f

∈ A[x1, . . . , xn] ai

∈ A

f

xαi

f ai · xαi ai = 0 f M(f )

f

m ∈ M(f ) deg(m) =

rk=0 αk m f ∈ A[x1, . . . , xn]

deg(f ) := max{deg(m) | m ∈ M(f )}

deg(f ) = 1

f

A[x1, . . . , xn]

α

A[x1, . . . , xn] ∩ A

f ∈ A[n]

deg(f ) < 1

f ∈ A

f

(A, +, ·)

(A[x], +, ·)

f + g =max{n,m}i=0 cix

i

ci = ai + bi n m bm+1 = bm+2 = · · · = bn = 0

n m

f · g = n+mi=0 cixi ci = j+k=i a jbk

(A[x1, . . . , xn], +, ·)

A[x1, . . . , xn] = A[x1, . . . , xn−1][xn] n ∈ N n ≥ 2

n = 1

A[x1, . . . , xn−1]

xn A[x1, . . . , xn−1]


7/48

A[x1, . . . , xn] = A[x1, . . . , xn−1][xn] f ∈ A[x1, . . . , xn]

f =mi=1

aixαi11 . . . x

αinn , ai ∈ A αij ∈ N ∪ {0},

i ∈ Nm j ∈ Nn bi = aixαi11 . . . xαi n−1n−1 ∈ A[x1, . . . , xn−1]

f =

mi=1 bix

αinn ∈ A[x1, . . . , xn−1][xn]

(A, +, ·)

A[x1, . . . , xn]

A[x1, . . . , xn] = A[x1, . . . , xn−1][xn] n ≥ 2

n = 1

f =r1

i=0 aixi1 =

0 g =

r2i=0 bix

i1 = 0 r1, r2 ∈ N ai, bi ∈ A ar1 , br2 = 0 f · g = 0

ar1br2 = 0

(A, +,

·)

A[x

1, . . . , x

n]

A

A ⊂ A[x1, . . . , xn] A

A[x1, . . . , xn] n = 1 f ∈ A[x1]

f /∈ A

deg(f ) ≥ 1

g ∈ A[x1] f · g = 1 A[x1] deg(1) = deg(f ) + deg(g) 1 deg(f ) 1 deg(g) 0

deg(1) < 1

A[x1, . . . , xn−1]

A

f ∈ A[x1, . . . , xn] \ A f ∈ A[x1, . . . , xn] \ A[x1, . . . , xn−1]

A[x1, . . . , xn−1] deg(1) 1

A[x1, . . . , xn] A

(A, +1, ·1) (B, +2, ·2) ψ : A → B

a, b ∈ A

ψ(a ·1 b) = ψ(a) ·2 ψ(b)

ψ(a +1 b) = ψ(a) +2 ψ(b) ψ(1A) = 1B Im(ψ) = {b ∈ B |

ψ(a) = b

a ∈ A}

ψ Ker(ψ) = {a ∈ A | ψ(a) = 0}

ψ

ψ

A

B

ψ : A → B Im(ψ) Ker(ψ)

B A

Ker(ψ)

A

c, d ∈ I m(ψ)

a, b ∈ A

ψ(a) = c

ψ(b) = d

cd = ψ(a)ψ(b) = ψ(ab) c − d = ψ(a) − ψ(b) = ψ(a − b),

ab ∈ A

a − b ∈ A

ψ(ab) = cd ∈ I m(ψ)

ψ(a − b) = a − b ∈ I m(ψ)

Im(ψ)

B

e, f ∈ K er(ψ)

ψ(e) = ψ(f ) = 0

ψ(e − f ) = ψ(e) − ψ(f ) = 0 ψ(ef ) = ψ(e)ψ(f ) = 0,

e − f,ef ∈ K er(ψ) Ker(ψ) A

a ∈ A b ∈ Ker(ψ)

ψ(ab) = ψ(a)ψ(b) = ψ(a)0 = 0

ab ∈ Ker(ψ)

Ker(ψ)

Ker(ψ)

A


8/48

A

I ⊂ A

A

I

A/I = {a + I | a ∈ A}

a + I = {a + i | i ∈ I }

a ∈ A

R

I ⊂ R

π : R → R/I

π(a) = a + I

a ∈ R

I

A

B

ψ : A → B

A/Ker(ψ)

Im(ψ)

ψ : A/Ker(ψ) → I m(ψ)

ψ(x + Ker(ψ)) = ψ(x)

∀ x ∈ A

ψ

a + Ker(ψ) = b + Ker(ψ), a , b ∈ A

a − b ∈ Ker(ψ)

ψ(a − b) = ψ(a) − ψ(b) = 0

ψ(a) = ψ(b)

ψ(a + Ker(ψ)) = ψ(b + K er(ψ))

b ∈ I m(ψ)

a ∈ A

a ∈ A

ψ(a) = b

b = ψ(a + Ker(ψ))

ψ(a + K er(ψ)) = ψ(b + K er(ψ))

ψ(a) = ψ(b)

a − b ∈ Ker(ψ)

a + Ker(ψ) = b + Ker(ψ)

K[n]

K

M = ∅

a a, ∀a ∈ M

a, b ∈ M

a b

b a

a = b

a,b,c ∈ M

a b

b c

a c

a ≺ b

a

b

b ≺ a

a = b

a, b ∈ M

a b

a

b

a,b, ∈ M

M

K[x1, . . . , xn]

M

M

a, b ∈ M

a b

a · c b · c, ∀c ∈ M


9/48

M

M

m, n ∈ M

m =

ni=0 xi

αi

n =n

j=0 x jβj

m L n m n γ = β − α

k ∈ N

γ p = 0 p < k γ k > 0

m LR n k ∈ N γ p = 0 k < p n γ k > 0

f ∈ K[x1, . . . , xn]

f

ml(f ) := max{m|m ∈ M(f )} f

cl(f )

ml(f )

f tl(f ) := ml(f ) · cl(f )

p, q ∈ K[x1, . . . , xn] p q q

p

p|q

t ∈ K[x1, . . . , xn] q = p · t

p, q ∈ M

p = x1a1 . . . xn

an

q =x1

b1 . . . xnbn

ai, bi 0 i ∈ Nn p q t ∈ M

t = x1c1 . . . xn

cn

ci = bi − ai 0 i ∈ Nn

f, g ∈ K[n]

g = 0

q, r ∈ K[n]

f = q · g + r ml(g) ml(r)

ml(g) ml(f )

q = 0

r = f f r = q = 0 ml(g) | ml(f ) f 1 = f q 1 = tl(f 1)/tl(g) f 2 = f 1 − q 1 · g tl(f 1)

f 2 ml(f 2) ml(f 1) (f n) f i+1 = f i − q i · g q i = tl(f i)/tl(g) q i ml(g) | ml(f i) f i = 0

f m

f m = f m−1−q m−1 ·g ml(g) ml(f m)

f m−1

f m = f m−1 − q m−1 · g = f m−2 − q m−2 · g − q m−1 · g

f m

−2, f m

−3, . . . , f 2 f = f 1 = (q 1+q 2+

· · ·+q m

−1)

·g +f m

q = (q 1 + q 2 + · · · + q m−1) r = f m f = q · g + r

q 1, q 2, r1, r2 ∈ K[n] f = q 1 · g + r1

f = q 2 · g + r2 ml(g) ml(r1) ml(g) ml(r2) ml(g) ml(r1)

ml(g) m

m ∈ M(r1) r2 q 1 · g + r1 = q 2 · g + r2 (q 1 − q 2) · g = r2 − r1 r2 = r1 tl(g) | tl(r2 − r1)

M(r2 − r1) ⊆ M(r1) ∪M(r2) r2 = r1 q 2 = q 1

r

f

g

rg(f ) q f g

f 1, . . . , f k ∈ K[n] f ∈ K[n]

a1, a2, . . . , ak r ∈ K[n] f = a1 · f 1 + ... + ak · f k + r ml(r) ml(f i)

i ∈ Nk


10/48

f ∈ K[n]

a1, r1 ∈ K[n]

f = f 1 · a1 + r1 tl(f 1) tl(r1) a2, r2 ∈ K[n] r1 = f 2 · a2 + r2 f = f 1 · a1 + f 2 · a2 + r2

ri f i

m > 0

tl(f k) tl(rm) k ∈ Nn f f = a1 · f 1 + a2 · f 2 + · · · + ak · f k + rm

r

f

G = {f 1, . . . , f k} rG(f )

f = xy2 − x, g1 = xy − y g2 = y2 − x ∈ K[x, y]

y ≺ x

f

g1 g2

f = yg1 + g2 g2 g1

f = xg2 + 0.g1 + x2 − x

x2 − x = 0

K[x1, . . . , xn]

(A, +, ·)

I ⊂ A

I

A

0

∈ I

a, b ∈ I

a + b ∈ I

a ∈ I

b ∈ A

a · b ∈ I

A

{0}

A

A

a1, . . . , as ⊂ A

A

a1, . . . , as

a1, a2, . . . , ak ∈ A a1, a2, . . . , ak

{a1 · b1 + a2 · b2 + · · · + ak · bk | b1, b2, . . . , bk ∈ A},

a1, a2, . . . , ak a1, a2, . . . , ak

a1, a2, . . . , ak

f 1, . . . , f k ∈ A f 1, . . . , f k A

0 = a1 · 0 + · · · + f k · 0 ∈ f 1, . . . , f k 0 ∈ A

a, b ∈ f 1, . . . , f k a1, . . . , ak, b1, . . . , bk

a = f 1 · a1 + · · · + f k · ak


11/48

b = f 1 · b1 + · · · + f k · bk.

a + b = f 1 · (a1 + b1) + · · · + f k · (ak + bk),

a + b ∈ f 1, . . . , f k c ∈ A

a · c = f 1 · a1 · c + · · · + f k · ak · c

ai · c ∈ A i ∈ Nk a · c ∈ f 1, . . . , f k f 1, . . . , f k A

I ∩ J

I

J

I + J

I, J

A

I ∩ J

I + J

A

I + J = {a + b | a ∈ I , b ∈ J }

I, J

0 ∈ I

0 ∈ J

0 ∈ I ∩ J

a, b ∈ I ∩ J a + b ∈ I a + b ∈ J a, b ∈ I a, b ∈ J I J a + b ∈ I ∩ J

c ∈ A

a · c ∈ I

a · c ∈ J

a · c ∈ I ∩ J

I ∩ J

a, b ∈ I + J

a1, b1 ∈ I a2, b2 ∈ J a = a1 + a2 b = b1 + b2 a + b = (a1 + a2) + (b1 + b2) = (a1 + b1) + (a2 + b2)

I

J

a1 + b1 ∈ I a2 + b2 ∈ J a + b ∈ I + J

a ∈ I + J

b ∈ A

a · b ∈ I + J

S ⊂ I

S = I

S

I

I

I

K[x1, . . . , xn]

I

A

L ⊂ I

L ⊂ I

L ⊂ M ⊂ A

L ⊂ M

a ∈ L a L

I

L ⊂ I

a ∈ I

L ⊂ I

p ∈ L

p

L

M

p ∈ M

L ⊂ M

I

A

S

I

S

I

I

G ={g1, . . . , gs} gi ∈ I, ∀i ∈ Ns S

gi S S i ⊂ S

gi X = ∪si=1S i G

I

X ⊂ S

I


12/48

K[x]

I = {0}

I = K[x]

f ∈ I

deg(f )

{deg(g) | g ∈ I }

g ∈ I

q g, rg ∈ K[x]

g = f · q g + rg tl(g) tl(rg) deg(f ) > deg(rg) rg ∈ I f, q g · g ∈ I

deg(f )

I

=

{0

} I

= K[x]

rg = 0 g

∈ f

K[n]

I ⊂ K[n]

I = {0}

I = {0}

K[x]

K[n

−1]

f 1 ∈ I f 1 = g1 · xa1n g1 ∈ K[n−1] a1 ∈ N

f 1 xn I = f 1

f 2 ∈ I \ f 1

f 2 = g2 ·xa2n g2 ∈ K[n−1] a2 ∈ N I = f 1, f 2

f r = grxarn ∈ I \ f 1, f 2 . . . , f r−1

r ∈ N

ar f r

u ∈ N

I = f 1, . . . , f u

G =

{g1, g2, . . .

}

J =

G

⊂ K[n−1]

M = {m1, . . . , ms} ⊂ J J = M

G

M

G

M

M

G

i ∈ N

j ∈ N

pi ∈ K[n−1]

gi = pi · m j j k ∈ N q j ∈ K[n−1] m j = q j · gk

gi = pi · q j · gk i = k i = k pi · q j = 1

pi ∈ K M G

M

G

M

f

G i > k

f i = gi · xain = pi · q j · gk · xain = pi · q j · gk · xai−akn · xakn = pi · q j · xai−akn · f k.

f i f k i > k f i

f 1, . . . , f k, . . . , f i−1 i < k

I

K[n]

f ∈ K[n]

I = m1, . . . , ms

M(f ) ⊂ I


13/48

f

I

f

f ∈ I

q 1, . . . , q s ∈ K[n] f = m1 · q 1 + · · · + ms · q s

ml(f ) = ml(m1 · q 1 + · · · + ms · q s) = m1 · q 1 ml(f ) ∈ I f 1 = f − tl(f ) f 1 ∈ I

f 1 f f

I

M(f ) ⊂ I


14/48

ml(I ) = {ml(f ) | f ∈ I }

I

I

in(I )

ml(I )

in(I ) = ml(I )

K[n]

I

K[n]

G

I

v ∈ ml(I )

u ∈ ml(G)

v

u

I ⊆ K[n]

G =

{g1, . . . , g p} ⊂ I

G

I

in(I ) = ml(G)

f ∈ I

rG(f ) = 0

f ∈ I

f =

si=1 q i · gi ml(f ) = max1is{ml(q i) ·

ml(gi)}

1 ⇒ 2

g ∈ ml(I )

m1, . . . , ms ∈

ml(I )

g = si=1 mi i ∈ Ns f i ∈ I hi ∈ K[n] mi = hi · ml(f i) G I

f i ∈ I u ∈ G ml(f i) ml(u) f i ∈ ml(G) g ∈ ml(G)

ml(I ) ⊂ ml(G)

G ⊂ I

ml(G) ⊂ ml(I )

ml(G) ⊂ ml(I )


15/48

2 ⇒ 3

f ∈ I

q 1, q 2, . . . q s, rG(f ) ∈ K[n]

f =

si=1 gi · q i + rG(f ) ml(rG(f )) ml(gi) i ∈ Ns

q =

si=1 gi · q i q G ⊂ I q ∈ I

f ∈ I

rG(f ) = f − q ∈ I ml(rG(f )) ∈ ml(I ) ⊂ ml(I ) = ml(G)

ml(rG(f )) gi rG(f ) = 0

rG(f ) = 0 f G I

3 ⇒ 4

f ∈ I

rG(f ) = 0

f

G

q 1, q 2 . . . , q s ∈ K[n] f = si=1 q i · gi ml(f ) = ml(si=1 q i · gi) =

max1is{ml(q i) · ml(gi)} 4 ⇒ 1

f ∈ I

ml(f ) = max1is{ml(q i) · ml(gi)}

ml(G)

G

I

K[n]

I

⊂K[n]

G =

{g1, . . . , gs

} I in(I ) = in(G) = ml(G)

G

I

in(I ) = ml(I ) = ml(G) = in(G) ⊂ in(G),

ml(G) ⊂ ml(G)

J = G J ⊂ I

G ⊂ I

ml(J ) ⊂ ml(I )

in(J ) ⊂ in(I )

in(G) = in(I )

G ⊂ G

ml(G) ⊂ ml(G)

ml(G)

⊂ ml(

G

)

= in(G

).

f ∈ in(G)

f

ml(G)

ml(I )

G ⊂ I

f

ml(I )

ml(I )

ml(G)

G

f

ml(G)

f ∈ ml(G)

in(G) = ml(G)

K[n]

I, J ⊆ K[n]

J ⊂ I

in(I ) = in(J )

I = J

I = J

I \ J = ∅

I \ J

f ∈ I \ J

tl(f )

f ∈ I ml(f ) ∈ ml(I )

ml(f ) ∈ in(I ) = in(J )

f ∈ J

in(I ) = in(J )

g ∈ J

tl(g) = tl(f )

tl(f − g) ≺ tl(f )

f

I \ J

f − g ∈ J

g ∈ J

(f − g) + g = f ∈ J

I = J

I ⊂ K[n]

G

I

G

I

G ⊂ I J = G ⊂ I in(I ) =ml(G) = in(J )

I = J

G

I

K[n]


16/48

K[n]

I ⊂ K[n]

J = in(I )

m1, . . . , ms ∈ J J = m1, . . . , ms

f ∈ I

ml(f ) ∈ ml(I ) ⊂ J

ml(f )

mi

G =

{f 1, . . . , f s

}

f i ml(f i) = mi, i

∈ Ns G

I

K[n]

G

I

m1 =

ni=1 x

aii , m2 =

ni=1 x

bii ∈

M

m1 m2 mmc(m1, m2) = ni=1 x

cii ci = max

{ai, bi

}

i ∈ Nn

M

f, g ∈ K[n]

f

g

S (f, g)

S (f, g) := mmc(ml(f ), ml(g)) · f tl(f )

− mmc(ml(f ), ml(g)) · gtl(g)

K[x, y]

y ≺ x

g1 =

xy − y g2 = y

2 − x

S (g1, g2) = (xy2) xy − y

xy − (xy2)y2 − x

y2 = y(xy − y) − x(y2 − x) = x2 − y2.

K[x1, . . . , xn]

f, g ∈ K[x1, . . . , xn]

S (f, g) = −S (g, f )

m1, m2 ∈ M S (m1, m2) = 0

ml(mmc(ml(f ), ml(g)) f tl(f )

) = mmc(ml(f ), ml(g))

ml(S (f, g)) ≺ mmc(ml(f ), ml(g))


17/48

S (f, g) = mmc(ml(f ), ml(g)) · ( f

tl(f ) − g

tl(g)) = −mmc(ml(f ), ml(g)) · ( g

tl(g) − f

tl(f )) = −S (g, f )

S (m1, m2) = mmc(ml(f ), ml(g)) · ( m1tl(m1) − m2tl(m2)) = mmc(ml(f ), ml(g)) · (m1m1 − m2m2 ) = 0

q = mmc(ml(f ),ml(g))

tl(f ) ml(f ) =

pi=1 xi

ai ml(g) =

pi=1 xi

bi

mmc(ml(f ), ml(g)) =

pi=1

ximax{ai,bi}.

ml(q · f ) = q · ml(f ) = pi=1 xiγ i γ i = (ci − ai) + ai = ci ml(q · f ) =

mmc(ml(f ), ml(g))

ml(S (f, g)) = ml(mmc(ml(f ), ml(g)) f

tl(f ) − mmc(ml(f ), ml(g)) g

tl(g)) ≺

≺ max{ml(f ), ml(g)} mmc(ml(f ), ml(g))

S (f, g)

ml(S (f, g))

ml(f )

ml(g)

K[n]

G = {g1, . . . , gs} I

rG(S (gi, g j)) = 0 gi, g j ∈ G

G

I

I

g1, g2

∈G S (g1, g2) ∈ I rG(S (g1, g2)) = 0

f ∈ I

G

G

g ∈ G

ml(g) | ml(f )

h1, h2, . . . , hs ∈ K[n] f = h1 · g1 + ... + hs · gs L = hl · gl = max(ml(hi · gi) |

i ∈ Ns) f L

L = ml(f )

gl | ml(f )

L = ml(f )

ml(f ) ≺ L

k ∈ Ns ml(hk · gk) = ml(f ) L H

f

L

H =

{i

| ml(hi

·gi) = L

}

f

H H L = ml(f )

f

H

h1 · g1 h2 · g2

h1 · g1 + h2 · g2 = tl(h1) · g1 + tl(h2) · g2 + (h1 − tl(h1)) · g1 + (h2 − tl(h2)) · g2

tl(h1 · g1) = tl(tl(h1) · g1) ml((h1 − tl(h1)) · g1) = ml(h1 · g1 − tl(h1) · g1) ≺ L,

tl(h1) · g1 + tl(h2) · g2 = tl(h1 · g1) · (tl(h1) · g1tl(h1 · g1) +

tl(h2) · g2tl(h2 · g2)).

tl(h1 · g1) = −tl(h2 · g2)


18/48

tl(h1) · g1 + tl(h2) · g2 = tl(h1 · g1) · ( tl(h1)·g1tl(h1·g1) + tl(h2)·g2tl(h2·g2)) =

= tl(h1 · g1) · ( g1tl(g1) + g2tl(g2)

) = tl(h1·g1)mmc(ml(g1),ml(g2))

· S (g1, g2).

rG(S (g1, g2)) = 0 S (g1, g2)

G

p1, . . . , ps ∈ K[n] f

G

f = tl(h1 · g1)

mmc(ml(g1), ml(g2))

si=1

pi · gi + (h1 − tl(h1)) · g1 + (h2 − tl(h2)) · g2 +s

i=3

hi · gi.

ml((h1 − tl(h1)) · g1) ≺ L ml((h2 − tl(h2)) · g2) ≺ L

ml(S (g1, g2) · ml(h1g1)mmc(ml(g1), ml(g2))

) ≺ L

L

h j ·g j f ml(h j ·g j) = L

ml(f ) ≺ L H

H

f

H

ml(f ) = L

G

I

G

G

G

G

I

K[n]

S ⊂ I

S = I

G

S

G = S

P = {(g, g) | g, g ∈ G, g = g }

P = ∅

(g, g) ∈ P

P

r = rG(S (g, g))

r = 0

G = G ∪ {r}

P = P ∪ {(g, r) | g ∈ G}

G

P = ∅

(g, g) ∈ G

G

G

K[n]

I i ⊂ K[n], i ∈N I i ⊆ I i+1, ∀i ∈

N

m ∈ N

I m = I m+ j j ∈ N


19/48

I = ∪i∈NI i 0 ∈ I i ⊆ I i 0 ∈ I f, g ∈ I

i

j

N

f ∈ I i g ∈ I j i = j I i

f + g ∈ I i ⇒ f + g ∈ I i = j i > j f ∈ I j I i ⊆ I j f + g ∈ I j ⊆ I f ∈ I a ∈ K[n]

i

f ∈ I i af ∈ I i ⊆ I I

f 1, . . . , f r ∈ I

f 1, . . . , f r

= I

f k I j f 1

∈ I j1, . . . , f r

∈ I jr

m = max{ jk | k ∈ Nr} {f 1, . . . , f r} ⊂ I m

f 1, . . . , f r = I ⊆ I m ⊆ I ,

j

I ⊆ I m ⊆ I m+ j ⊆ I I m = I m+ j j ∈ N

G

G

r1, r2, . . .

G1 = G Gi+1 = Gi ∪ {ri} i ∈ N I i = Gi I i ⊂ I i+1 Gi ⊂ Gi+1

i ∈ N

d ∈ N

j ∈ N

I j = I j+d G j = G j+d

G j G j I

P = ∅

P

y ≺ x

K[x, y]

S = {g1 = xy−y, g2 =

y2 − x}

I = S

f = xy2 − x ∈ I

rS (f ) S

I

S (g1, g2) = (xy2) xy − yxy − (xy2)y2

− xy2 = y(xy − y) − x(y2 − x) = x2 − y2,

rS (S (g1, g2)) = x

2 + x = 0

g3 = x2 + x

G = {g1, g2, g3} G

• S (g1, g2) = g3 rG(S (g1, g2)) = 0 • S (g1, g3) = (x2y)xy−yxy −(x2y)x

2−xx2

= x(xy−y)−y(x2−x) = −xy+yx = 0

rG(S (g1, g3)) = 0

• S (g2, g3) = (x2y2)x2−xx2 − (x2y2)y2−xy2

= y2(x2 − x) − x2(y2 − x) = x3 − xy2 = xg3 − xg2 rG(S (g2, g3)) = 0

G

I

G

I ⊂ K[n]

G ⊂ I

I

g, h ∈ G

ml(g) | ml(h)

H = G \ {h}

I


20/48

f ∈ I

g, h ∈ G

ml(g) | ml(h)

G

p ∈ G

ml( p) | ml(f )

p ∈ H

p = h

ml(g) | ml(h)

ml( p) = ml(h) | ml(f )

ml(g) | ml(f )

g ∈ H

H

I

I ⊂ K[n]

G ⊂ I

I

G

G

g1, g2 ∈ G ml(g1) ml(g2)

G = {g1 = x · y − y, g2 = y2 − x, g3 = x2 + x}

I = g1, g2 g1, g2 g3 tl(g1) = x · y

tl(g2) = x

2

tl(g3) = x2

G

I ⊂ K[n]

G ⊂ I

H ⊂ I

I

G

H

G

H

ml(hi) = ml(gi) i = 1, . . . , #G

G

H

#G = s < t = #H

g1 ∈ ml(G) h1 h2 ml(H )

h3 ml(H ) g1 h1 h2

h3 = h1 = h2 #G = #H

g1 ∈ G h1 ∈ H ml(g1) | ml(h1) H

ml(h1) | ml(g1) h ∈ H

ml(g1) g ∈ G

ml(g) | ml(h) | ml(g1) ml(g1) = ml(h1)

g2 ∈ G g2 = g1

G

H

I ⊂ K[n]

G ⊂ I

I

G

G

g ∈ G

h ∈ G \ {g}

rh(g) = g

G

G

G \ {g}


21/48

I

K[n]

G

I

P = {ml(g) + rG(g − ml(g)) | g ∈ G}

I

G

G \ {g}

g ∈ G

g

H = G \ {g}

G

g

rH (g) G m

∈N

H

m

−1

g1, g2, . . . , gm

−1

q 1, . . . , q m−1 ∈ K[n]

g =m−1i=1

gi · q i + rH (g),

rH (g) = ml(g) +rH (g −ml(g))

g

rH (g) = g −m−1i=1

giq i = ml(g) + rH (g − ml(g)),

I

H

H

rH (g −ml(g)) G

ml(g)

Q = {ml(g) + rH (g − ml(g)) | g ∈ H },

Q

I

g rH (g − ml(g)) Q = P

P

I

G = {g1 = x ·y −y, g2 = y2−x, g3 = x2+x}

I = g1, g2

gi G gi

K[n]

I ⊂ K[n]

I

G = {g1, . . . , gk} H =

{h1, . . . , hk}

ml(hi) = ml(gi), i ∈ Nk

gi ∈ G gi G gi

hi H gi = hi

gi ≺ hi rH (gi) = 0 M(gi) ml(H )

H

ml(hi) gi ≺ hi

ml(h p) ml(g j) ml(gi) rgj (gi) = gi j = i G j = i

ml(g j) = ml(gi) | ml(h p) | ml(gi) ⇒ ml(hi) = ml(gi) = ml(h p),

ml(h p) = ml(hi) i ∈ Nk

gi = hi H = G

K[n]


22/48

I, J ⊆ K[n]

G

J = G = I

C[x,y,z ]

z ≺ y ≺ x

x2 − yz = 3,y2 − xz = 4,z 2 − xy − 5.

S = {x2 − yz − 3, y2 − xz − 4, z 2 − xy − 5}

13x + 11z = 0,13y − z = 0,36z 2 = 169.

z

f ∈ K[n]

f : Kn → K

p ∈Kn

f

f ( p) = 0

q ∈ Kn

C

C[z ]

C

C[n]

f ∈ C[x1, . . . , xn] degxn(f ) > 1 f xn)

x1, . . . , xn−1 xn C

x1, . . . , xn−1

C[n]

p ∈ C[n]

q ∈ C[n]

p · q = 1


23/48

p

deg( p) < 1

p ∈ C

q = 1

p

q · p = 1

p

p

C

p

q

p · q = 1

p

u ∈ Cn

p(u) = 0

p(z ) · q (z ) = 1

z ∈ Cn

C[n]

z ∈ Cn, p(z ) = 0

q (z ) = 0

u

p

C[n]

p1, p2, . . . , pm ∈ C[n] q 1, . . . , q m ∈ C[n] p1 · q 1 + · · · + pm · q m = 1

S p

s1, . . . , sr ∈ K[n] I = s1, . . . , sr G = {g1, . . . , gt}

G

G p S p

z ∈ Kn

si(z ) = 0 i ∈ Nr S p g

I

g =

ri=1 ai · si ai ∈ K[n] g(z ) =

ri=1 ai(z ) · si(z ) = 0

z

G p G p

S p

I ⊂ K[x1, . . . , xn] G I

I j = I ∩K[x1, . . . , xn− j], j ∈ N

I

G j = G ∩K[x1, . . . , xn− j]

I j

K[x, y]

y ≺ x

G = {g1 = x · y − y, g2 = y2 − x, g3 = x2 − x} g1, g2

{g1, g2} g3 = 0 x = 0

x = 1

g2 (0, 0), (1, −1) (1, 1)

g1

S = {x2 −

yz − 3, y2 − xz − 4, z 2 − xy − 5}

C[x,y,z ]

z ≺ y ≺ x

13x + 11z = 0,13y − z = 0,36z 2 = 169.


24/48

z

z = 13

6 z = −13

6

13x = −11 · (±13

6 ),

13y = ±136 .

(−

11

6

, 1

6

, 13

6

) (11

6

,−

1

6

,−

13

6

)

C

R

C

f, f 1, f 2, . . . , f m ∈ C[x1, . . . , xn] f 1, . . . , f m

f

s ∈ N \ {0}

f s ∈ I = f 1, . . . , f m

y

g = 1 − yf ∈ C[x1, . . . , xn, y] J =

f 1, . . . , f m, g C[x1, . . . , xn] ⊂C[x1, . . . , xn, y] f

C = {f 1, . . . , f m} T = C ∪ {g}

α = (α1, . . . , αn, αy) ∈ Cn+1 f 1(α) = f 2(α) =

· · · = f m(α) = g(α) = 0 f (α) = 0 1 − αyf (α) = g(α) = 1

1 ∈ T

h, h1, . . . , hm ∈ C[x1, . . . , xn, y]

1 = h(1 − yf ) + h1f 1 + · · · + hmf m

y = 1/f

1 =mi=1

hi(x1, . . . , xn, 1/f )f i(x1, . . . , xn, 1/f ).

s ∈ N \ 0

f s

g1, . . . , gm ∈ C[x1, . . . , xn]

f s =

mi=1

gi(x1, . . . , xn)f i(x1, . . . , xn),

f s ∈ I

f 1, . . . , f m f

s

f

f

f 1, . . . , f m

1 ∈ f 1, . . . , f m, 1 − yf

C[x1, . . . , xn] xn ≺ · · · ≺ x1 f 1, . . . , f m ∈

C[x1, . . . , xn]

γ i ∈ N \ {0} xγ ii ∈ ml(I ) i ∈ Nn I = f 1, . . . , f m


25/48

i ∈ Nn γ i ∈ N \ {0} xγ ii ∈ ml(I )

gn ∈ I ml(gn) = xγ nn gn ∈ C[xn]

gn = 0 degxn(gn) = γ n

gn xn gn−1 ∈ I ml(gn−1) = xγ n−1n−1

gn−1 ∈ C[xn−1, xn] xn ≺ xn−1 z n gn = 0 gn−1(xn−1, z n) = 0 γ n−1 gn = gn−1 = 0

γ nγ n−1 i = 1, 2, . . . , n

−2

g1 = g2 = · · · = gn = 0 γ 1γ 2 . . . γ n

f 1 = f 2 = · · · = f m = 0 g1 = g2 = · · · = gn = 0

f 1, . . . , f m

l1 = (z 11, . . . , z n1), . . . , lk =(z 1k, . . . , z nk) hi =

k j=1(xi − z ij) ∈ C[xi] ⊂ C[x1, . . . , xn] i ∈ Nn k ∈ N

ml(hi) = x

ki i l1, . . . , lk hi i ∈ Nn

s1, . . . , sn {hs11 , . . . , hsnn } ⊂ I ml(hi) = xksii ∈ ml(I )

i ∈ Nn

K

n ∈ N

Kn = {(a1, . . . , an) |

ai ∈ K, i ∈ Nn} n

f 1, . . . , f n ∈ K[n] I ⊆ K[n]

I

V (I ) = {a ∈ Kn | f (a) = 0, ∀f ∈ I }

I ⊂ K[n]

Kn

I

V (I )

I

W

I (W ) = {f ∈ K[n] | f (a) = 0, ∀a ∈ W }

I ⊆ I (V (I ))

I (V (I )) = I

I (V (I ))

I

I

G

J

G

x1, x2, . . . , xn B

J = G

C[x1, . . . , xn]

G = 0

G

G

V (J )

f

G

x1, x2, . . . , xn


26/48

R[n]

f, g1, g2, . . . , g p, h1, . . . , hq ∈ R[n] n,p,q ∈ N

f : Rn → R

gi : Rn → R

i ∈ N p h j : Rn → R j ∈ Nq

f (x1, . . . , xn)

gi 0, i ∈ N ph j = 0, j ∈ Nq.

gi hi

I ⊂ R

f : I → R

z ∈ I

L ∈ R

f (x)

x

z

ε > 0

δ > 0

0


27/48

I ⊂ R

f : I → R

z ∈ I

f

z

limx→z

f (x) − f (z )x − z .

f

z

f (z )

∂f (z)∂x

f : I

→R

I f f

∂ ∂x

f

f

∂ 2f ∂x2

n

f (n)

∂ nf ∂xn

∂ ∂x

f, g : I ⊂ R → R

a, b ∈ R

∂ (a · f + b · g)

∂x

= a

·

∂f

∂x

+ b

·

∂g

∂x

.

f (x) = xn

f (x) =

nxn−1 ∀n ∈ N

f (x) = an · xn + · · · + a1 · x + a0 f (x)

f (x) = n · an · xn−1 + · · · + 2 · a2 · x + a1.

f, g : R → R

Im(g) ⊆Dom(f )

(f (g(x))) = f (g(x)) · g(x)

f : I ⊂ R → R

a ∈ I

f

V

a

f (a) f (x)

x ∈ V

x = a

a

I

f : I ⊂ R → R

f

a ∈ I

∂f (a)∂x

= 0

a

f : I ⊂ R → R

∂ 2f (a)∂x2

> 0

a

∂ 2f (a)∂x2

< 0

∂ 2f (a)∂x2

= 0

bases de gröbner e aplicações em otimização

Documents