Difference between revisions of "Package alggeozd"

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== Package Discription ==
 
== Package Discription ==
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The <code>zerodim</code> package provides functions for computing the introduced invariants of zero-dimensional schemes. In the graded case the package also provides functions for computations of the Hilbert functions of these invariants. Alias of the package is <code>ZD</code>.
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=== List of main functions ===
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[[/MinQuotIdeal/]]
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<pre>
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MinQuotIdeal(P, I, J): computes a min. homog. system
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          of generators of homog. ideal (I+J)/I.
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    input: P=K[x[1..N]], I and J homog. ideals of P
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    output: list of polys
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</pre>
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=== Example for computations ===
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Now let us apply the <code>zerodim</code> package to some concrete examples. Recall that the alias of the package is <code>ZD</code>, and so to call a function from this package in computation one uses [[/ZD.functions-name/]].
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Consider the first example, where X is the scheme defined by the homogeneous ideal Ix.
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<pre>
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Use P ::= QQ[X[0..2]];
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Ix := ideal(X[0]*X[1] -X[1]^2, X[1]^2*X[2] -X[1]*X[2]^2, 2*X[0]^2*X[2] -3*X[0]*X[2]^2 +X[2]^3);
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</pre>
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Then we calculate the differents of X as follows:
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<pre>
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[[Category:Package zerodim]]
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[[Category:Package alggeozd]]
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[[Category:ApCoCoA Packages]]

Revision as of 21:37, 17 November 2022

This page describes the alggeozd package. The package contains various functions for checking algebraic and geometric property of zero-dimensional schemes and related computations. For a complete list of functions, see Category:Package alggeozd.


Algebraic and Geometric Properties

Let be a field,

Package Discription

The zerodim package provides functions for computing the introduced invariants of zero-dimensional schemes. In the graded case the package also provides functions for computations of the Hilbert functions of these invariants. Alias of the package is ZD.

List of main functions

MinQuotIdeal

MinQuotIdeal(P, I, J): computes a min. homog. system
          of generators of homog. ideal (I+J)/I.
     input: P=K[x[1..N]], I and J homog. ideals of P
     output: list of polys


Example for computations

Now let us apply the zerodim package to some concrete examples. Recall that the alias of the package is ZD, and so to call a function from this package in computation one uses ZD.functions-name.

Consider the first example, where X is the scheme defined by the homogeneous ideal Ix.

Use P ::= QQ[X[0..2]];
Ix := ideal(X[0]*X[1] -X[1]^2, X[1]^2*X[2] -X[1]*X[2]^2, 2*X[0]^2*X[2] -3*X[0]*X[2]^2 +X[2]^3);

Then we calculate the differents of X as follows: