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2022-10-19 20:09:07 +0100 marked best answer Rewriting linear combination of Groebner basis in terms of original terms

Let assume I have an ideal given by

x,y,z = QQ['x,y,z'].gens()
I = ideal(f1,f2,f3)
B = I.groebner_basis()

where f1,f2,f3 are just polynomials in variables x,y,z. Let's say B=(g1,g2).

Let's assume I happen to take a polynomial,h, that is in my ideal I. Then doing polynomial division, I can write

h=h1*g1+h2*g2

Basically I can write h as a linear combination of the elements in my Groebner basis. Is there a function that converts a linear combination in terms of Groebner to linear combination of terms in my ideal I? i.e.I can write

h=q1*f1+q2*f2+q3*f3
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2022-04-06 05:57:25 +0100 edited question Determining whether module is free for polynomial/power series rings

Determining whether module is free for polynomial/power series rings Say I have two modules R = QQ[[x, y, z]] S = QQ[[x

2022-04-04 18:24:23 +0100 commented question Determining whether module is free for polynomial/power series rings

Hello, I just made that as an example. So to make thing simpler, let's ignore the +1. If we think of polynomial ring fir

2022-04-04 18:23:17 +0100 commented question Determining whether module is free for polynomial/power series rings

Hello, I just made that as an example. So to make thing simpler, let's ignore the +1. If we think of polynomial ring fir

2022-04-02 06:50:20 +0100 asked a question Determining whether module is free for polynomial/power series rings

Determining whether module is free for polynomial/power series rings Hello, I am wondering if I have say two modules R=Q

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2021-12-07 02:13:18 +0100 asked a question What does the e in the output mean?

What does the e in the output mean? Hello, what does the e in the output mean? What number is -1.36424205265939e-12? See

2021-10-20 18:29:47 +0100 commented question Indenting in Cocalc vs Sage Notebook

Okay cool thanks.

2021-10-20 14:03:25 +0100 asked a question Indenting in Cocalc vs Sage Notebook

Indenting in Cocalc vs Sage Notebook Hello I started using Cocalc. I had a Sage Notebook installed on my computer. On my

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2021-07-31 17:00:20 +0100 marked best answer Solutions to Equations over different fields

Hello, suppose I want to solve a system of equations

w,x,y,z = SR.var('w, x, y, z')
eqn1=2*w*y^2 + y^2*z + z^3==0
eqn2=4*x^3 - 2*x*y^2 - y^3==0
eqn3=2*w^2*y - 2*x^2*y - 3*x*y^2 + 2*w*y*z==0
eqn4=w*y^2 + 3*w*z^2==0
sol = solve([eqn1, eqn2, eqn3, eqn4], w,x,y,z, solution_dict=True)
sol

This gives me solutions over complex numbers I believe? How can I ask for solutions over say a finite field. Let's make it more simple and say the field is a prime and not a prime power. Hence, just reducing modulo p. As you all know, it is possible more solutions arise when moving to finite fields.

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2021-07-30 17:50:25 +0100 commented answer Solutions to Equations over different fields

Hello take p=5, what would it mean if the dimension of variety is still one after adding in the Frobenius equations? For

2021-07-30 17:48:28 +0100 commented answer Solutions to Equations over different fields

Hello, what would it mean if the dimension of variety is still one after adding in the Frobenius equations? For example,

2021-07-29 06:30:15 +0100 asked a question Solutions to Equations over different fields

Solutions to Equations over different fields Hello, suppose I want to solve a system of equations w,x,y,z = SR.var('w,

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2021-06-03 07:32:08 +0100 commented answer Multivariate Polynomial Ring +1 variable

Great thanks.

2021-06-03 07:31:58 +0100 commented answer Multivariate Polynomial Ring +1 variable

Great thanks.

2021-06-03 07:31:50 +0100 marked best answer Multivariate Polynomial Ring +1 variable

So the idea is I was first working over

R.<w,x,y,z>=QQ[]

I have a function f and J is the Jacobian of f belonging to the ring above. I do some stuff and I end with a polynomial g in a symbolic ring in variables w,x,z. I want to lift g. So I want to do

q1, q2, q3, q4 = g.lift(J)
q1=1/3 * q1(w,x,y,z).derivative(w)
q2=1/3 *q2(w,x,y,z).derivative(x)
q3=1/3 *q3(w,x,y,z).derivative(y)
q4=1/3*q4(w,x,y,z).derivative(z)
h1=q1+q2+q3+q4

Now, the Symbolic ring has no attribute lift. This can be fixed by moving to Multivariate Polynomiial Ring by doing

g=g.polynomial(QQ)

The issue is, because g is only a function in w,x,z, this moves g to the Multivariate Polynomial Ring of w,x,z. This gives error as the Jacobian and function f is in Multivariate Polynomial Ring of w,x,y,z. I want g to be in the Multivariate Polynomial Ring of w,x,y,z even though there is no y in g. How can I do this? See my 2 attachment. In the attachment, h2 plays the role of g in my explanation above. C:\fakepath\Screenshot (126).pngC:\fakepath\Screenshot (123).png

2021-06-02 22:26:07 +0100 edited question Multivariate Polynomial Ring +1 variable

Multivariate Polynomial Ring +1 variable So the idea is I was first working over R.<w,x,y,z>=QQ[] I have a fun

2021-06-02 22:25:51 +0100 edited question Multivariate Polynomial Ring +1 variable

Multivariate Polynomial Ring +1 variable So the idea is I was first working over R.<w,x,y,z>=QQ[] I have a fun

2021-06-02 22:25:02 +0100 edited question Multivariate Polynomial Ring +1 variable

Multivariate Polynomial Ring +1 variable So the idea is I was first working over R.<w,x,y,z>=QQ[] I have a fun

2021-06-02 22:23:39 +0100 asked a question Multivariate Polynomial Ring +1 variable

Multivariate Polynomial Ring +1 variable So the idea is I was first working over R.<w,x,y,z>=QQ[] I have a fun

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2021-05-13 02:54:44 +0100 asked a question Singularity Type

Singularity Type Hello, so in Singular, there is a way to give an affine equation and determine the singularity type. Fo

2021-05-11 03:48:34 +0100 commented answer Determinant Function on Qp

Great thanks.

2021-05-11 03:48:24 +0100 marked best answer Determinant Function on Qp

See attachmentC:\fakepath\Screenshot (104).png

I basically have 2 matrices with entries in Qp. I cant exactly do Matrix2.charpoly() as this is det (xI-Matrix2) and I want determinant of (I-xMatrix2). So I decided to just type det(I-x*Matrix2) but it gives me error. Is there a way to fix this?

If you want me to be honest, I get an error just from I-x*Matrix2. So it might not even be the determinant function.

2021-05-10 23:25:41 +0100 commented answer Determinant Function on Qp

Great thanks. I have a question on this. Suppose I have a matrix with rational entries. So let's say Matrix is = [a, b]