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Questions tagged [macaulay2]

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

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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
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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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Hilbert polynomials in two variables with Macaulay2

In J. Symb. Comput. (1999) 28, 681-710, Levin worked with bifiltered, finitely generated $R$-modules ($R$ being a polynomial ring in two sets of variables) and he found an analogue of the Hilbert ...
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Using the `initialIdeal` function in Macaulay2

My understanding is that there's an initialIdeal function in Macaulay2 for computing intial ideals with respect to a grading, specifically in the ...
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1answer
52 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
37 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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57 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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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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1answer
186 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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1answer
293 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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37 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
118 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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1answer
229 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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228 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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53 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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228 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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1answer
407 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
110 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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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
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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
226 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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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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1answer
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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
437 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
467 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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91 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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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
343 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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286 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 $I$...