The Structure of Polynomial Ideals And Grobner Bases
The Structure of Polynomial Ideals And Grobner Bases
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4 PARTITIONING THE POLYNOMIAL SPACE 10 Lemma 9 Let h e A'[ii, . . . , i„+i] be a homogeneous polynomial. Then a{HeadB{h)) = Jea is defined such that the monomial whose a projection is >-least will appear as HeadB(/i). Q. E. D.
Theorem 10 Lei G = {dii- ■ 'iSk} be a homogeneous Grdhner basis for I relative to >. Then, G = {cr{g\), - ■ ■ ■, cr{gk)} is a Grobner basis for I B relative to >.
A Proof. By the previous lemma, we know that G is a basis for /, so we only need to prove that G is Grob...ner. We use the characterization that G is a Grobner basis if and only if (Head^(G)) = Headyi{I). Clearly Head^(G) C Head^(7), so we need to show the other inclusion.
Since G is a basis for 7, any /i G / can be written as h = IIaicr(g, ) with a, e K[xi, . .. , !„]. Let h' = Efliffn then a(/j') = h.
Now h' G /, and / is a homogeneous ideal. Write h' = Hf-o h'^ where h'^ is the degree z homogeneous component of h'.
What to read after The Structure of Polynomial Ideals And Grobner Bases?
You can find similar books in the "Read Also" column, or choose other free books by Thomas Dube to read onlineMoreLess
To quickly assess the difficulty of the text, read a short excerpt:
4 PARTITIONING THE POLYNOMIAL SPACE 10 Lemma 9 Let h e A'[ii, . . . , i„+i] be a homogeneous polynomial. Then a{HeadB{h)) = Jea is defined such that the monomial whose a projection is >-least will appear as HeadB(/i). Q. E. D.
Theorem 10 Lei G = {dii- ■ 'iSk} be a homogeneous Grdhner basis for I relative to >. Then, G = {cr{g\), - ■ ■ ■, cr{gk)} is a Grobner basis for I B relative to >.
A Proof. By the previous lemma, we know that G is a basis for /, so we only need to prove that G is Grob...ner. We use the characterization that G is a Grobner basis if and only if (Head^(G)) = Headyi{I). Clearly Head^(G) C Head^(7), so we need to show the other inclusion.
Since G is a basis for 7, any /i G / can be written as h = IIaicr(g, ) with a, e K[xi, . .. , !„]. Let h' = Efliffn then a(/j') = h.
Now h' G /, and / is a homogeneous ideal. Write h' = Hf-o h'^ where h'^ is the degree z homogeneous component of h'.
What to read after The Structure of Polynomial Ideals And Grobner Bases?
You can find similar books in the "Read Also" column, or choose other free books by Thomas Dube to read onlineMoreLess
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