groebner basis 3
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Grobner Bases Introduction 3
Daniel Cabarcas, University of Cincinnati
Cryptography SeminarCincinnati, April 20, 2009
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Construction of Grobner bases
Theorem (characterization of Grobner bases)G = {g1, . . . , gt} I is a Grobner basis of I w.r.t. a monomialorder iffFor all f K[ x] there exist unique r K[ x] and g I s.t.
f = g + r
and no monomial of r is divisible by any of LM(g1), . . . , LM(gt).
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Construction of Grobner bases
Theorem (characterization of Grobner bases)G = {g1, . . . , gt} I is a Grobner basis of I w.r.t. a monomialorder iffFor all f K[ x] there exist unique r K[ x] and g I s.t.
f = g + r
and no monomial of r is divisible by any of LM(g1), . . . , LM(gt).
The crucial idea in Grobner basis theory is the
observation that these infinitely many tests can bereplaced by the consideration of finitely many critical
situations that can be characterized by the so-called
S-polynomials. Bruno Buchberger 1999
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Construction of Grobner basis
LT(g1), . . . , LT(gt) = LT(I)
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Construction of Grobner basis
LT(g1), . . . , LT(gt) = LT(I)
Let f, g K[ x]
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Construction of Grobner basis
LT(g1), . . . , LT(gt) = LT(I)
Let f, g K[ x]
let m = lcm(LM(f), LM(g))
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Construction of Grobner basis
LT(g1), . . . , LT(gt) = LT(I)
Let f, g K[ x]
let m = lcm(LM(f), LM(g))
The s-polynomial of f and g is
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Construction of Grobner basis
LT(g1), . . . , LT(gt) = LT(I)
Let f, g K[ x]
let m = lcm(LM(f), LM(g))
The s-polynomial of f and g is
S(f, g) :=m
LT(f) fm
LT(g) g
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Construction of Grobner basis
LT(g1), . . . , LT(gt) = LT(I)
S(f, g) :=m
LT(f)f
m
LT(g)g
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Construction of Grobner basis
LT(g1
), . . . , LT(gt) = LT(I)
S(f, g) :=m
LT(f)f
m
LT(g)g
e.g. S(x2y 1, xy2 1)
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Construction of Grobner basis
LT(g1
), . . . , LT(gt) = LT(I)
S(f, g) :=m
LT(f)f
m
LT(g)g
e.g. S(x2y 1, xy2 1)
=x2y2
x2y(x2y 1)
x2y2
xy2(xy2 1)
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Construction of Grobner basis
LT(g1
), . . . , LT(gt) = LT(I)
S(f, g) :=m
LT(f)f
m
LT(g)g
e.g. S(x2y 1, xy2 1)
=x2y2
x2y(x2y 1)
x2y2
xy2(xy2 1)
=y(x2y 1) x(xy2 1)
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Construction of Grobner basis
LT(g1
), . . . ,
LT(gt) = LT(I)
S(f, g) :=m
LT(f)f
m
LT(g)g
e.g. S(x2y 1, xy2 1)
=x2y2
x2y(x2y 1)
x2y2
xy2(xy2 1)
=y(x2y 1) x(xy2 1)
=(x2y2 y) (x2y2 x)
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Construction of Grobner basis
LT
(g1
), . . . , LT
(gt) =
LT(
I)
S(f, g) :=m
LT(f)f
m
LT(g)g
e.g. S(x2y 1, xy2 1)
=x2y2
x2y(x2y 1)
x2y2
xy2(xy2 1)
=y(x2y 1) x(xy2 1)
=(x2y2 y) (x2y2 x)
=x y
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Construction of Grobner basis
TheoremLet I be an ideal of K[ x]. Then a set of generatorsG = {g1, . . . , gt} for I is a Grobner basis for I iff for all i = j
S(gi, gj)
G
0
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G
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Construction of Grobner basis
TheoremLet I be an ideal of K[ x]. Then a set of generatorsG = {g1, . . . , gt} for I is a Grobner basis for I iff for all i = j
S(gi, gj)
G
0
LemmaLet f 1, . . . , fs K[ x] and c1, . . . , cs K. Suppose that forLM(f1) = LM(f2) = = LM(fs) =: m and that
LM(
si=1 ci fi) < m.
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Construction of Grobner basis
TheoremLet I be an ideal of K[ x]. Then a set of generatorsG = {g1, . . . , gt} for I is a Grobner basis for I iff for all i = j
S(gi, gj)
G
0
LemmaLet f 1, . . . , fs K[ x] and c1, . . . , cs K. Suppose that forLM(f1) = LM(f2) = = LM(fs) =: m and that
LM(
si=1 ci fi) < m.Then
si=1 ci fi is a linear combination with
coefficients in K of the s-polynomials S(fi, fj) for 1 j < k s.
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Construction of Grobner basis
TheoremLet I be an ideal of K[ x]. Then a set of generatorsG = {g1, . . . , gt} for I is a Grobner basis for I iff for all i = j
S(gi, gj)
G
0
LemmaLet f 1, . . . , fs K[ x] and c1, . . . , cs K. Suppose that forLM(f1) = LM(f2) = = LM(fs) =: m and that
LM(
si=1 ci fi) < m.Then
si=1 ci fi is a linear combination with
coefficients in K of the s-polynomials S(fi, fj) for 1 j < k s.Furthermore, LM(S(fi, fj)) < m
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Construction of Grobner basis
TheoremLet I = f1, . . . , fs be a polynomial ideal. Then a Grobner basisfor I can be constructed in a finite number of steps by the
following algorithm:
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Construction of Grobner basis
TheoremLet I = f1, . . . , fs be a polynomial ideal. Then a Grobner basisfor I can be constructed in a finite number of steps by the
following algorithm:
Buchberger algorithm(F = {f1, . . . , fs})
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Construction of Grobner basis
TheoremLet I = f1, . . . , fs be a polynomial ideal. Then a Grobner basisfor I can be constructed in a finite number of steps by the
following algorithm:
Buchberger algorithm(F = {f1, . . . , fs})
G := F
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Construction of Grobner basis
TheoremLet I = f1, . . . , fs be a polynomial ideal. Then a Grobner basisfor I can be constructed in a finite number of steps by the
following algorithm:
Buchberger algorithm(F = {f1, . . . , fs})
G := F
REPEAT
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Construction of Grobner basis
TheoremLet I = f1, . . . , fs be a polynomial ideal. Then a Grobner basisfor I can be constructed in a finite number of steps by the
following algorithm:
Buchberger algorithm(F = {f1, . . . , fs})
G := F
REPEAT G := G
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Construction of Grobner basis
TheoremLet I = f1, . . . , fs be a polynomial ideal. Then a Grobner basisfor I can be constructed in a finite number of steps by the
following algorithm:
Buchberger algorithm(F = {f1, . . . , fs})
G := F
REPEAT G := G FOR EACH pair {f, g} f = g in G
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Construction of Grobner basis
TheoremLet I = f1, . . . , fs be a polynomial ideal. Then a Grobner basisfor I can be constructed in a finite number of steps by the
following algorithm:
Buchberger algorithm(F = {f1, . . . , fs})
G := F
REPEAT G := G FOR EACH pair {f, g} f = g in G
Compute S(f,
g)
G r
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Construction of Grobner basis
TheoremLet I = f1, . . . , fs be a polynomial ideal. Then a Grobner basisfor I can be constructed in a finite number of steps by the
following algorithm:
Buchberger algorithm(F = {f1, . . . , fs})
G := F
REPEAT G := G FOR EACH pair {f, g} f = g in G
Compute S(f,
g)
G r IF r = 0 THEN G := G {r}
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Construction of Grobner basis
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Construction of Grobner basis
TheoremLet I = f1, . . . , fs be a polynomial ideal. Then a Grobner basisfor I can be constructed in a finite number of steps by the
following algorithm:
Buchberger algorithm(F = {f1, . . . , fs})
G := F
REPEAT G := G FOR EACH pair {f, g} f = g in G
Compute S(f,
g)
G r IF r = 0 THEN G := G {r}
UNTIL G = G
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