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2023-03-26 22:09:33 +0100 | asked a question | How to solve for coefficients in a Quotient ring? How to solve for coefficients in a Quotient ring? I'm finding that the solve function doesn't take into account the quot |
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2021-09-14 06:49:48 +0100 | asked a question | How to access magma interface within sage when path is broken? How to access magma interface within sage when path is broken? I am trying to call magma from within sage, and am not ab |
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2021-04-26 00:48:13 +0100 | edited answer | How to express the coefficients of an iterated group action on a basis as a matrix? Just in case this is helpful for anyone else, here is the functional code answering my original question: R.<y1, y2 |
2021-04-26 00:41:09 +0100 | answered a question | How to express the coefficients of an iterated group action on a basis as a matrix? Just in case this is helpful for anyone else, here is the functional code answering my original question: R.<y1, y2 |
2021-04-26 00:23:14 +0100 | marked best answer | How to express the coefficients of an iterated group action on a basis as a matrix? I have a basis $(b_i)_{i \in I} = (1, y_1, y_2, y_1^2, y_1y_2, y_2^2, y_1^2y_2, y_1y_2^2, y_1^2y_2^2)$, and a group action $g$ on that basis, where $g$ acts by: $g(y_1) = y_1+1$, and $g(y_2) = y_2-y_1^2-y_1$. For each basis element $b_i$, I want the array $(b_i, g(b_i), g^2(b_i), ..., g^8(b_i))$. I want to see the dimension of the vector space spanned by those values. My first and foremost question is as follows: How can I get sage to put the coefficients of the array of polynomials (in a quotient ring) into columns of a matrix? I can't seem to get sage to spit out the coefficients at all! I tried Here is my code for context: Secondly, how can I more concisely run the iteration itself? Here is how I implemented an iterated group action on a basis element. I list only the $b_4$ example for readability, the others are the same but with $b_i$ Edit: I also realized that for $b_3$, the output of iteration has $y_1^3$ in it, even though we are considering its image in the quotient ring. How could this be happening? |
2021-04-24 23:39:51 +0100 | edited question | How to express the coefficients of an iterated group action on a basis as a matrix? How to express the coefficients of an iterated group action on a basis as a matrix? I have a basis $(b_i)_{i \in I} = (1 |
2021-04-24 23:39:47 +0100 | edited question | How to express the coefficients of an iterated group action on a basis as a matrix? How to express the coefficients of an iterated group action on a basis as a matrix? I have a basis $(b_i)_{i \in I} = (1 |
2021-04-24 23:33:18 +0100 | commented answer | How to express the coefficients of an iterated group action on a basis as a matrix? If I understand correctly, your code outputs one matrix: (g(e)) for each e in E, that is: (g(1), g(y1), g(y1^2), ...., g |
2021-04-24 23:30:29 +0100 | commented answer | How to express the coefficients of an iterated group action on a basis as a matrix? If I understand correctly, your code outputs (g(e)) for each e in E, that is: (g(1), g(y1), g(y1^2), ...., g(y1^2y2^2)), |
2021-04-24 22:16:43 +0100 | edited question | How to express the coefficients of an iterated group action on a basis as a matrix? How to express the coefficients of an iterated group action on a basis as a matrix? I have a basis $(b_i)_{i \in I} = (1 |
2021-04-24 22:16:30 +0100 | edited question | How to express the coefficients of an iterated group action on a basis as a matrix? How to express the coefficients of an iterated group action on a basis as a matrix? I have a basis $(b_i)_{i \in I} = (1 |
2021-04-24 22:13:36 +0100 | marked best answer | Why am I getting a type error when I attempt to take the projective closure of this intersection? I am attempting to take the projective closure of the intersection of the following affine polynomials (not the intersection of the closure!): $$y^3-y-x^2= 0 $$ $$w^3-w+y^7-y^5-x^4y^3+x^4y = 0.$$ This affine intersection is a curve of dimension one. Unfortunately, I haven't been able to enter this lovely affine curve defined by this intersection into sage. When I attempt as follows: I get an error at the definition of C due to the second polynomial: I am so confused because this is absolutely in the coordinate ring of $A$. Why am I getting this type error? How can I enter this affine intersection into sage, so that I may take its closure? As an aside, I can enter the intersection of the closure, which is not what I want, as follows: The intersection of the closure has an extra irreducible component $[x: 0: w: 0]$, |
2021-04-24 21:56:38 +0100 | asked a question | How to express the coefficients of an iterated group action on a basis as a matrix? How to express the coefficients of an iterated group action on a basis as a matrix? I have a basis $(b_i)_{i \in I} = (1 |
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2021-04-01 05:08:59 +0100 | asked a question | Why am I getting a type error when I attempt to take the projective closure of this intersection? Why am I getting a type error when I attempt to take the projective closure of this intersection? I am attempting to tak |
2021-02-07 20:01:27 +0100 | marked best answer | How can I compose 2 power series in one variable with their compositional inverse get a power series in two variables? I would like to compose a power series $\ell$ defined in $x$ to get a power series $\ell^{-1}(\ell(x) + \ell(y))$ as a power series $f(x, y)$ in two variables, $x$ and $y$. In the code below I call l := $\ell$, and e := $\ell^{-1}$. I find immediately the following issue, let alone the issue of composing: Once I have this two variable power series $f(x, y)$, I would like to output $f(x, (f(x, ..., f(x,x)))$, composed with itself $n$-times for a natural number $n$. |
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2021-02-07 20:00:38 +0100 | commented answer | Dedekind Zeta function of cyclotomic field wrongly evaluating to zero on -1? So the value of KL(-1) is indeed zero. This is very troubling! Thank you. |
2021-02-06 23:44:49 +0100 | asked a question | Dedekind Zeta function of cyclotomic field wrongly evaluating to zero on -1? Let $K := \mathbb{Q}(\zeta)$ be the pth cyclotomic extension of $\mathbb{Q}$. I would like to verify the results of a paper which states the quotient of their Dedekind zeta functions have particular values. Below Z is the Riemann zeta function (Dedekind zeta function of $\mathbb{Q}$). The expected values are nonzero! For example.
Here is my first question: Have I incorrectly implemented the Dedekind Zeta function of a cyclotomic number field? Why is Here is my second question: How do I implement the evaluation of the L-series after I've taken their quotient? That is, |
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2020-09-14 03:46:36 +0100 | asked a question | How to implement Hensel's Lemma recursion? I am trying to automate a Hensellian lift calculation. In particular, I am stuck on implementing the ** step in a for-loop, or plugging a function into itself, as eval doesn't work to type change from string to integer when you are using it to define a variable. Does anyone know how to get around this or have a suggestion? I tried to instead use a seperate function, but the same issue arises. But again, it isn't quite right, ack! I don't know how to handle the recursion and I'm banging my head on the wall, it must be simple but I keep getting it slightly wrong. Thank you very much for your time. |
2020-04-24 03:03:07 +0100 | marked best answer | How to truncate a power series in two variables? I would like to truncate power-series by setting $y$ to 0, in order to express $y = x^3 - xy^2$ as a power series in $x$ by recursively plugging in the equation for $y$, then truncating. I have found that other variants, such as |
2020-04-24 03:00:38 +0100 | asked a question | How to base change from a PolynomialRing to that Ring with one variable evaluated, i.e., from Q[x,y] to Q[x,y]/(x=0) = Q[y]? I am trying to base change a Laurent series ring element from its base ring, Q[u1, u2, u3], to a quotient of its base ring, Q[u2, u3], but I am quite confused in forming this quotient. My setting is this: In other words, I wish to set u1 = 0, and look at f over that ring. I tried the following two things, which spit out f unchanged. I also tried the following which gives an attribute error: I am completely stuck and would deeply appreciate any help. I would also like to eventually set u2 = 0 and look at f over that ring, which I mention at the off chance that this changes the answer at all. |
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2020-04-19 20:54:05 +0100 | marked best answer | Why is sage not recognizing symmetric polynomials as symmetric? I am trying to decompose symmetric polynomials into polynomial combinations of elementary symmetric polynomials. It has been driving me absolutely up the wall, and I would be very grateful for any help. It works perfectly with 2 variables. For 3 variables it fails. Sage throws a type error and says the polynomial f is not symmetric, even though it is certainly invariant under the action of the symmetric group of order 3. The type error is as folliows: However, the following does not throw a type error, perhaps because it is a linear combination of the elementary basis rather than a polynomial one. I don't understand what is going on. Why is this type error being triggered for a valid symmetric polynomial? Thank you very much for your time. |