How is “binomial” defined in Algebraic Geometry?
0
I am learning ideal arithmetics and I was flabbergasted that $langle xrangle$ is binomial ideal, as observed with Macaulay2 here. $x$ is clearly not a polynomial with two terms. Then I read this paper:
A binomial is a polynomial with at most two terms; a binomial ideal is an ideal generated by binomials.
So in the context of Algebraic geometry, how can I differentiate the two different meanings about binomial? When does "binomial" mean two and when does "binomial" mean at most two? Why is such difference made?
algebraic-geometry polynomials terminology ideals
edited Dec 8 at 23:46
Rodrigo de Azevedo
12.8k41855
asked Apr 11 '16 at 21:56
hhh
2,80753476
|
show 1 more comment
0
I am learning ideal arithmetics and I was flabbergasted that $langle xrangle$ is binomial ideal, as observed with Macaulay2 here. $x$ is clearly not a polynomial with two terms. Then I read this paper:
A binomial is a polynomial with at most two terms; a binomial ideal is an ideal generated by binomials.
So in the context of Algebraic geometry, how can I differentiate the two different meanings about binomial? When does "binomial" mean two and when does "binomial" mean at most two? Why is such difference made?
algebraic-geometry polynomials terminology ideals
edited Dec 8 at 23:46
Rodrigo de Azevedo
12.8k41855
asked Apr 11 '16 at 21:56
hhh
2,80753476
2
I'm not sure this is the answer since I have not seen this terminology before, but $langle xrangle = langle x+x^2,x-x^2rangle$ so it satisfies the definition.
– Captain Lama
Apr 11 '16 at 21:59
@CaptainLama good point, still it does not explain "A binomial is a polynomial with at most two terms". I would call $x$ monomial, not binomial. Yet now I understand the "at most" for binomial ideal, thank you.
– hhh
Apr 11 '16 at 22:20
1
What do you mean it does not explain ? $x+x^2$ and $x-x^2$ are binomials according to this definition, so $langle x+x^2,x-x^2rangle$ is generated by binomials. It's like saying that $langle x, x+y^2rangle$ is not a homogeneous ideal because $x+y^z$ is not homogeneous : this ideal is also $langle x,y^2rangle$ so it is homogeneous. You just have to find the right generators.
– Captain Lama
Apr 11 '16 at 22:23
1
Yes, I think basically there are two (related) notions here : binomial polynomials, and binomial ideals. And the example of $langle xrangle$ shows that a principal ideal $langle Prangle$ can be a binomial ideal even when $P$ is not a binomial polynomial (which is different from homogeneous ideals, where $langle Prangle$ is homogeneous iff $P$ is a homogenous polynomial).
– Captain Lama
Apr 11 '16 at 22:32
1
Ah, yes this clearly seems like a bug, you should do a bug report.
– Captain Lama
Apr 11 '16 at 22:40
|
show 1 more comment
0
0
0
1
I am learning ideal arithmetics and I was flabbergasted that $langle xrangle$ is binomial ideal, as observed with Macaulay2 here. $x$ is clearly not a polynomial with two terms. Then I read this paper:
A binomial is a polynomial with at most two terms; a binomial ideal is an ideal generated by binomials.
So in the context of Algebraic geometry, how can I differentiate the two different meanings about binomial? When does "binomial" mean two and when does "binomial" mean at most two? Why is such difference made?
algebraic-geometry polynomials terminology ideals
edited Dec 8 at 23:46
Rodrigo de Azevedo
12.8k41855
asked Apr 11 '16 at 21:56
hhh
2,80753476
I am learning ideal arithmetics and I was flabbergasted that $langle xrangle$ is binomial ideal, as observed with Macaulay2 here. $x$ is clearly not a polynomial with two terms. Then I read this paper:
A binomial is a polynomial with at most two terms; a binomial ideal is an ideal generated by binomials.
So in the context of Algebraic geometry, how can I differentiate the two different meanings about binomial? When does "binomial" mean two and when does "binomial" mean at most two? Why is such difference made?
algebraic-geometry polynomials terminology ideals
algebraic-geometry polynomials terminology ideals
edited Dec 8 at 23:46
Rodrigo de Azevedo
12.8k41855
asked Apr 11 '16 at 21:56
hhh
2,80753476
edited Dec 8 at 23:46
Rodrigo de Azevedo
12.8k41855
asked Apr 11 '16 at 21:56
hhh
2,80753476
edited Dec 8 at 23:46
Rodrigo de Azevedo
12.8k41855
edited Dec 8 at 23:46
Rodrigo de Azevedo
12.8k41855
edited Dec 8 at 23:46
Rodrigo de Azevedo
12.8k41855
12.8k41855
asked Apr 11 '16 at 21:56
hhh
2,80753476
asked Apr 11 '16 at 21:56
hhh
2,80753476
asked Apr 11 '16 at 21:56
hhh
2,80753476
2,80753476
2
I'm not sure this is the answer since I have not seen this terminology before, but $langle xrangle = langle x+x^2,x-x^2rangle$ so it satisfies the definition.
– Captain Lama
Apr 11 '16 at 21:59
@CaptainLama good point, still it does not explain "A binomial is a polynomial with at most two terms". I would call $x$ monomial, not binomial. Yet now I understand the "at most" for binomial ideal, thank you.
– hhh
Apr 11 '16 at 22:20
1
What do you mean it does not explain ? $x+x^2$ and $x-x^2$ are binomials according to this definition, so $langle x+x^2,x-x^2rangle$ is generated by binomials. It's like saying that $langle x, x+y^2rangle$ is not a homogeneous ideal because $x+y^z$ is not homogeneous : this ideal is also $langle x,y^2rangle$ so it is homogeneous. You just have to find the right generators.
– Captain Lama
Apr 11 '16 at 22:23
1
Yes, I think basically there are two (related) notions here : binomial polynomials, and binomial ideals. And the example of $langle xrangle$ shows that a principal ideal $langle Prangle$ can be a binomial ideal even when $P$ is not a binomial polynomial (which is different from homogeneous ideals, where $langle Prangle$ is homogeneous iff $P$ is a homogenous polynomial).
– Captain Lama
Apr 11 '16 at 22:32
1
Ah, yes this clearly seems like a bug, you should do a bug report.
– Captain Lama
Apr 11 '16 at 22:40
|
show 1 more comment
2
I'm not sure this is the answer since I have not seen this terminology before, but $langle xrangle = langle x+x^2,x-x^2rangle$ so it satisfies the definition.
– Captain Lama
Apr 11 '16 at 21:59
@CaptainLama good point, still it does not explain "A binomial is a polynomial with at most two terms". I would call $x$ monomial, not binomial. Yet now I understand the "at most" for binomial ideal, thank you.
– hhh
Apr 11 '16 at 22:20
1
What do you mean it does not explain ? $x+x^2$ and $x-x^2$ are binomials according to this definition, so $langle x+x^2,x-x^2rangle$ is generated by binomials. It's like saying that $langle x, x+y^2rangle$ is not a homogeneous ideal because $x+y^z$ is not homogeneous : this ideal is also $langle x,y^2rangle$ so it is homogeneous. You just have to find the right generators.
– Captain Lama
Apr 11 '16 at 22:23
1
Yes, I think basically there are two (related) notions here : binomial polynomials, and binomial ideals. And the example of $langle xrangle$ shows that a principal ideal $langle Prangle$ can be a binomial ideal even when $P$ is not a binomial polynomial (which is different from homogeneous ideals, where $langle Prangle$ is homogeneous iff $P$ is a homogenous polynomial).
– Captain Lama
Apr 11 '16 at 22:32
1
Ah, yes this clearly seems like a bug, you should do a bug report.
– Captain Lama
Apr 11 '16 at 22:40
2
2
I'm not sure this is the answer since I have not seen this terminology before, but $langle xrangle = langle x+x^2,x-x^2rangle$ so it satisfies the definition.
– Captain Lama
Apr 11 '16 at 21:59
I'm not sure this is the answer since I have not seen this terminology before, but $langle xrangle = langle x+x^2,x-x^2rangle$ so it satisfies the definition.
– Captain Lama
Apr 11 '16 at 21:59
@CaptainLama good point, still it does not explain "A binomial is a polynomial with at most two terms". I would call $x$ monomial, not binomial. Yet now I understand the "at most" for binomial ideal, thank you.
– hhh
Apr 11 '16 at 22:20
@CaptainLama good point, still it does not explain "A binomial is a polynomial with at most two terms". I would call $x$ monomial, not binomial. Yet now I understand the "at most" for binomial ideal, thank you.
– hhh
Apr 11 '16 at 22:20
1
1
What do you mean it does not explain ? $x+x^2$ and $x-x^2$ are binomials according to this definition, so $langle x+x^2,x-x^2rangle$ is generated by binomials. It's like saying that $langle x, x+y^2rangle$ is not a homogeneous ideal because $x+y^z$ is not homogeneous : this ideal is also $langle x,y^2rangle$ so it is homogeneous. You just have to find the right generators.
– Captain Lama
Apr 11 '16 at 22:23
What do you mean it does not explain ? $x+x^2$ and $x-x^2$ are binomials according to this definition, so $langle x+x^2,x-x^2rangle$ is generated by binomials. It's like saying that $langle x, x+y^2rangle$ is not a homogeneous ideal because $x+y^z$ is not homogeneous : this ideal is also $langle x,y^2rangle$ so it is homogeneous. You just have to find the right generators.
– Captain Lama
Apr 11 '16 at 22:23
1
1
Yes, I think basically there are two (related) notions here : binomial polynomials, and binomial ideals. And the example of $langle xrangle$ shows that a principal ideal $langle Prangle$ can be a binomial ideal even when $P$ is not a binomial polynomial (which is different from homogeneous ideals, where $langle Prangle$ is homogeneous iff $P$ is a homogenous polynomial).
– Captain Lama
Apr 11 '16 at 22:32
Yes, I think basically there are two (related) notions here : binomial polynomials, and binomial ideals. And the example of $langle xrangle$ shows that a principal ideal $langle Prangle$ can be a binomial ideal even when $P$ is not a binomial polynomial (which is different from homogeneous ideals, where $langle Prangle$ is homogeneous iff $P$ is a homogenous polynomial).
– Captain Lama
Apr 11 '16 at 22:32
1
1
Ah, yes this clearly seems like a bug, you should do a bug report.
– Captain Lama
Apr 11 '16 at 22:40
Ah, yes this clearly seems like a bug, you should do a bug report.
– Captain Lama
Apr 11 '16 at 22:40
|
show 1 more comment
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2
I'm not sure this is the answer since I have not seen this terminology before, but $langle xrangle = langle x+x^2,x-x^2rangle$ so it satisfies the definition.
– Captain Lama
Apr 11 '16 at 21:59
@CaptainLama good point, still it does not explain "A binomial is a polynomial with at most two terms". I would call $x$ monomial, not binomial. Yet now I understand the "at most" for binomial ideal, thank you.
– hhh
Apr 11 '16 at 22:20
1
What do you mean it does not explain ? $x+x^2$ and $x-x^2$ are binomials according to this definition, so $langle x+x^2,x-x^2rangle$ is generated by binomials. It's like saying that $langle x, x+y^2rangle$ is not a homogeneous ideal because $x+y^z$ is not homogeneous : this ideal is also $langle x,y^2rangle$ so it is homogeneous. You just have to find the right generators.
– Captain Lama
Apr 11 '16 at 22:23
1
Yes, I think basically there are two (related) notions here : binomial polynomials, and binomial ideals. And the example of $langle xrangle$ shows that a principal ideal $langle Prangle$ can be a binomial ideal even when $P$ is not a binomial polynomial (which is different from homogeneous ideals, where $langle Prangle$ is homogeneous iff $P$ is a homogenous polynomial).
– Captain Lama
Apr 11 '16 at 22:32
1
Ah, yes this clearly seems like a bug, you should do a bug report.
– Captain Lama
Apr 11 '16 at 22:40