Questions tagged [macaulay2]

Macaulay2 is a software system devoted to supporting research in algebraic geometry and commutative algebra.

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How can I get maximal ideal containing an ideal using Macaulay2?

In Macaulay2, I have written the following codes to find the maximal ideal in the ring $Q[x,y,z]$ containing the ideal generated by $x^2y+z$ and $xz-y$. R=QQ[x,y,z] I=ideal(x^2y+z,xz-y) M=getMaxIdeal ...
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Macaulay2 Monomial Order

In Macaulay2, one can define a polynomial ring with a certain monomial order as follows: R=QQ[x,y,z,MonomialOrder=>{Lex=>2,Position=>Up}] This means $R$ is ...
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87 views

How to say a variable is invertible in Macaulay2?

I'm a very beginner in Macaulay2, so I apologize if this question is too trivial... I'm using Macaulay2 for a computation involving over $30$ variables. Roughly speaking I have a $4\times 4$ matrix ...
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39 views

Localization of a Weyl-Algebra module

My aim is to understand how to describe the localization of modules over the Weyl algebra. I want to be able to do simple examples by hand. I wrote the following code in Macaulay2. ...
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30 views

Twisted cubic curve; finding a representation as an ideal of minors of a 2 by 3 matrix of linear forms

Looking at a particular twisted cubic curve I was playing with getting the representation as the 2 by 2 minors of a 2 by 3 matrix of linear forms. See e.g. G. Ellingsrud, R. Piene and S.A. Strømme On ...
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31 views

Finding the length of the kernel of a map in Macaulay2

Given an equigenerated monomial ideal $I$ over a polynomial ring, I am trying to check if a sequence $L={l_1,\ldots,l_t}$ is almost $I$-regular, i.e. for each $i$, the kernel of the multiplication map ...
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62 views

Bhargava Higher Composition Laws I Computations

as a project for a class I'm currently taking, I've decided to undertake the execution of the proofs in Manjul Bhargava's expositional article, "Higher composition laws I: A new view on Gauss ...
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49 views

Create list with a for loop on Macaulay2

I'm trying to write a function on Macaulay2 such that, given a square polynomial matrix in input, put out a list $\{(1,p_1(x)), \ldots , (r,p_r(x))\}$ where $r$is the rank of the matrix and $p_i(x)$ ...
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54 views

Macaulay Matrix Reduction in Macaulay2

I am trying to get the Groebner bases of an ideal in Macaulay 2 through the triangularization of the Macaulay matrix. Indeed, I would like to know how far I should go -how big the matrix should be- ...
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which mehod can easily design index of Grassmannian and its k and n or any function for this in Macaulay2 and how to convert poset to this index?

how to know k and n and its index of Grassmannian? which mehod can easily design index of Grassmannian and its k and n or any library or function for this in Macaulay2? is there any library or ...
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241 views

Macaulay2: How to compute the remainder when dividing a polynomial by a set of polynomials (in some order)?

I'm writing Buchberger's Criterion in a program in Macaulay2 to check whether or not the set of polynomials I have form a Grobner basis for the ideal it generates. However, I have not been able to ...
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49 views

Confusion about length of module over different rings and Macaulay2 code about computing length of module.

I was trying to compute some examples dealing with length of modules and got stuck with this simple example: Let $R=k[t]/(t^2)$ where $k$ is a field and $J=(x^3)$ be the ideal of the polynomial ring $...
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54 views

Product of List of Polynomials in Macaulay2

I am currently writing a function in Macaulay2 which given an elementary symmetric function $e_i$ outputs its expression in terms of power sums according the formula: $$e_n=\sum_{|\lambda|=n}(-1)^{|\...
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66 views

Macaulay2 stuck analyzing `simple' ideal

I'm attempting to use Macaulay2 to compute minimal prime decompositions of various ideals. I'm getting used to the program and I've been successful in simpler cases. But with the ideal I present ...
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1answer
58 views

Use Macaulay2 to compute minimal primes of complicated ideal

I tried computing the minimal primes of a fairly complex ideal using the online Macauley2 interface. I start by letting R=QQ[z] and ...
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93 views

Ideal of $V(Z-XY,Y^2+XZ-X^2)$

Let $k$ be an algebraically closed field, not necessarily of characteristic $0$. I’m trying to compute the ideal of $C=V(Z-XY,Y^2+XZ-X^2)$ over $k$. Here is what I have tried so far: We have $$C=V(...
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How to compute rational points on a projective variety in Macaulay2

I know that the package "rationalPoints" will compute rational points on an affine variety over a finite field, but I would like to do the same for varieties in projective space. Is there a built-in ...
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1answer
52 views

Display results in Macaulay2 using text.

I use Macaulay2 to compute Hilbert series. But the results in Macaulay2 are of the following form: ...
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75 views

Find ideal of Grassmannian in Macaulay 2

This question is a technical question about M2. I know that in Macaulay 2 I can use "Grassmannian$(l,k)$" to get ideal of $(l+1)$-plane in $\mathbb{C}^{k+1}$ and the result ideal is in $\mathbb{ZZ}[...
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19 views

Finding complementary of a collection in Macaulay2

Having a collection A of sets of the same cardinality t, I am trying to find all elements in 2^[n]\A of cardinality t. Is it possible to do so in Macaulay2? If so, which packages are required? How ...
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298 views

Ring of formal power series in Macaulay2

How does one define the ring $\mathbb{C}[[x,y]]$ of formal power series in two variables over $\mathbb{C}$ in Macaulay2? Or Singular? I have seen some papers that claim to perform calculations in ...
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472 views

Can Macaulay2 do computations with symbolic parameters?

I'm trying to figure out how to use Macaulay2 to do some ideal membership computations, and I'm running into a problem with symbolic parameters. Here is a practical example. Consider the family of ...
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103 views

Decomposing an ideal using Macaulay2

I give Macaulay2 the ideal $I=(y^2, x) \in Q[x , y]$ and then I put decompose I. The result is $(x , y)$ but I do not understand why. Does it mean that $I = (x , y)$...
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Open cones in Macaulay2

I need to create some polyhedral cones and form their intersections. Macaulay2 looks like a great tool for this, and so I set up M2 on a Linux box. I've been able to make the cones and form ...
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1answer
260 views

What does Hilbert series of monomial ideals describe?

I am trying to understand the point of Hilbert series of monomial ideals. I am confused because Macaulay has commands for hilbertSeries, hilbertPolynomial and hilbertFunction. What does Hilbert ...
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333 views

Compute Ext with Macaulay2

I want to compute Ext with Macaulay2. I see in the website they write how to do it, but I can not do it. Can anyone help me with an example? For example, let $S=k[x,y,z,t]$. How to compute $\mathrm{...
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280 views

Singular CAS vs Macaulay2 for finite fields

I intend to work on error correcting codes using finite fields. Finite fields are supported by both Singular and Macaulay2. I am confused about which one I should start with to learn. Any suggestions ...
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60 views

Solving polynomial systems with homotopy. Where is the bottleneck?

I have a system of polynomial equations with $n$ unknowns (where $n$ can be between 3 and 20) and that is known to have at least $n!$ isolated solutions. I want to solve this system numerically, but ...
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2answers
373 views

Is a function in an ideal? Verification by hand and Macaulay 2

Suppose $$f_1=-4x^4y^2z^2+y^6+3z^5,$$ $$f_2=-4x^2y^2z^2+y^6+3z^5,$$ $$f_3=4x^4y^2z^2+y^6+3z^5,$$ $$f_4=4x^2y^2z^2+y^6+3z^5$$ and $$I=\langle xz-y^2,x^3-z^2\rangle\subset\mathbb C[x,y,z].$$ Is $...
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573 views

Calculating syzygies with Macaulay2

I'm trying to calculate the syzygies of a set of elements on the polinomial ring of 6 variables. But I'm trying to specify the number of generator in each degree the syzygies have. I know that ...
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1answer
164 views

Computing extensions of an ideal in Singular or Macaulay2

Does Macaulay2 or Singular compute extensions of ideals under ring homomorphisms? Specifically, if $\phi : R \to S$ is a ring homomorphism (say polynomial rings over $\mathbb{Q}$ which can be ...
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160 views

How to find all integral elements over a subring using Macaulay2?

I have the following question about Macaulay2. How to find all integral elements over a subring? What I mean is the following. Suppose $A$ is a subring of $B$. How can I find the following set? $$L=...
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1answer
115 views

Help with computation and Gröbner basis

I am learning a new software and a new topic (Gröbner basis). I have the following system of polynomial equations $$\begin{align} 6-21(x_1x_2+x_1x_3+x_1x_4) &= 0 \\ 10-21(x_2x_1+x_2x_3+x_2x_4) &...
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1answer
328 views

Finding the number of solutions of a system of equations in Macaulay 2

I just started working with Macaulay 2 and need some help. I need to find the number of solutions of a system of equations. I am having difficulty imputing this into the software so please be specific ...
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161 views

Using Macaulay 2 to find free divisors.

Given a hypersurface $D = h^{-1}(0)$ for some polynomial $h \in \mathbb{C} [x,y,z]$ I want to be able to use Macaulay 2 to tell if it's a free divisor or not. What I've got so far; Let $h_{p}$ be ...
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2k views

How can I get Macaulay2 to tell me if this ideal is prime?

I am trying to get Macaulay2 to confirm if $(y+zi,x^2 - z^2 - 1)$ is a prime ideal in $\Bbb{C}[x,y,z]$. Now as a small test, I tried to compute its radical by doing ...
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2answers
491 views

Radical ideal computation (Macaulay2)

Is there a way to find the radical ideal of $I_i=(a^n-u^{n-i+1}v^{n-i}, b^n-u^{i-1}v^i, uv-ab)$ for $1\leq i \leq n$ in $\mathbb{C}[u,v,a,b]?$ This is the generalization of my question here where I ...
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1answer
649 views

Radical of an ideal using Macaulay2 software.

What is the radical ideal of $(u^2v-a^3,uv^2-b^3,uv-ab)$ in $\mathbb{C}[u,v,a,b]?$ Above all, to learn how to fish, what would be code that I can use to get the radical? I have not worked with ...
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96 views

How to make such a matrix multiplication as fast as possible in Macaulay2?

Given two matrices, $A$ ($m$ rows and $n$ columns) and $B$ ($n$ rows and $k$ columns), we want to compute matrix $A$ acting on each row of matrix $B$, and expect $mk$-dimensional matrix $C$, namely $$...
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271 views

Primary decomposition of large ideals

Short version: I'd like to do a primary decomposition of an ideal with 38 generators in a polynomial ring with 44 generators. However, the ideal seems far too large to naively decompose in, say, ...
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1answer
519 views

Computing Betti numbers using Macaulay2

Let $k$ be a field and $R=k[x,y,z]$, let $M=R/\langle x^2,xy,yz^2,y^4\rangle$ be $R$-module, how can we compute the left free resolution of $M$, and also the Betti numbers of this resolution?
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315 views

Computing contractions of ideals in Macaulay2

Does Macaulay2 compute contractions of ideals under ring homomorphisms. Specifically, if $R\subseteq S$ is a ring extension (say polynomial rings over $\mathbb{Q}$ which can be specified in M2) and $I$...