<abstract xmlns="http://www.w3.org/1999/xhtml">An irreducible algebraic decomposition 
<inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/auom-2014-0042_i1.jpg"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><msubsup><mo>∪</mo><mrow><mi>i</mi><mo>=</mo><mn>0</mn></mrow><mi>d</mi></msubsup><msub><mi>X</mi><mi>i</mi></msub><mo>=</mo><msubsup><mo>∪</mo><mrow><mi>i</mi><mo>=</mo><mn>0</mn></mrow><mi>d</mi></msubsup><mrow><mo stretchy="false">(</mo><mrow><msubsup><mo>∪</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mrow><msub><mi>d</mi><mi>i</mi></msub></mrow></msubsup><msub><mi>X</mi><mi mathvariant="italic">ij</mi></msub></mrow><mo stretchy="false">)</mo></mrow></math><tex-math>$ \cup _{i = 0}^d X_i = \cup _{i = 0}^d ({\cup _{j = 1}^{d_i } X_{ij} } )$</tex-math></alternatives></inline-formula>
 of an affine algebraic variety <italic>X</italic> can be represented as a union of finite disjoint sets 
<inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/auom-2014-0042_i2.jpg"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><msubsup><mo>∪</mo><mrow><mi>i</mi><mo>=</mo><mn>0</mn></mrow><mi>d</mi></msubsup><msub><mi>W</mi><mi>i</mi></msub><mo>=</mo><msubsup><mo>∪</mo><mrow><mi>i</mi><mo>=</mo><mn>0</mn></mrow><mi>d</mi></msubsup><mrow><mo stretchy="false">(</mo><mrow><msubsup><mo>∪</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mrow><msub><mi>d</mi><mi>i</mi></msub></mrow></msubsup><msub><mi>W</mi><mi mathvariant="italic">ij</mi></msub></mrow><mo stretchy="false">)</mo></mrow></math><tex-math>$\cup _{i = 0}^d W_i = \cup _{i = 0}^d ({\cup _{j = 1}^{d_i } W_{ij} } )$</tex-math></alternatives></inline-formula>
 called numerical irreducible decomposition (cf. [14],[15],[18],[19],[20],[22],[23],[24]). The <italic>Wi</italic> correspond to the pure i-dimensional components <italic>Xi</italic>, and the <italic>Wij</italic> present the i-dimensional irreducible components <italic>Xij</italic>. The numerical irreducible decomposition is implemented in B<sc>ertini</sc> (cf. [3]). The algorithms use homotopy continuation methods. We modify this concept using partially Gröbner bases, triangular sets, local dimension, and the so-called zero sum relation. We present in this paper the corresponding algorithms and their implementations in S<sc>ingular</sc> (cf. [8]). We give some examples and timings, which show that the modified algorithms are more efficient if the number of variables is not too large. For a large number of variables B<sc>ertini</sc> is more efficient*.</abstract>

An irreducible algebraic decomposition 
∪i=0dXi=∪i=0d(∪j=1diXij)$ \cup _{i = 0}^d X_i = \cup _{i = 0}^d ({\cup _{j = 1}^{d_i } X_{ij} } )$
 of an affine algebraic variety X can be represented as a union of finite disjoint sets 
∪i=0dWi=∪i=0d(∪j=1diWij)$\cup _{i = 0}^d W_i = \cup _{i = 0}^d ({\cup _{j = 1}^{d_i } W_{ij} } )$
 called numerical irreducible decomposition (cf. [14],[15],[18],[19],[20],[22],[23],[24]). The Wi correspond to the pure i-dimensional components Xi, and the Wij present the i-dimensional irreducible components Xij. The numerical irreducible decomposition is implemented in Bertini (cf. [3]). The algorithms use homotopy continuation methods. We modify this concept using partially Gröbner bases, triangular sets, local dimension, and the so-called zero sum relation. We present in this paper the corresponding algorithms and their implementations in Singular (cf. [8]). We give some examples and timings, which show that the modified algorithms are more efficient if the number of variables is not too large. For a large number of variables Bertini is more efficient*.

Numerical Decomposition of Affine Algebraic Varieties

Department of Mathematics, University of Damascus, Syria.

Department of Mathematics, University of Kaiserslautern, Germany.

Analele ştiinţifice ale Universităţii "Ovidius" Constanţa. Seria Matematică

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

{"article-title":"Numerical Decomposition of Affine Algebraic Varieties"}

An irreducible algebraic decomposition 
∪i=0dXi=∪i=0d(∪j=1diXij)$ \cup _{i = 0}^d X_i = \cup _{i = 0}^d ({\cup _{j = 1}^{d_i } X_{ij} } )$
 of an affine...

Numerical Decomposition of Affine Algebraic Varieties

Published Online: Oct 20, 2015

Page range: 193 - 216

Received: Sep 01, 2012

Accepted: Feb 01, 2013

DOI: https://doi.org/10.2478/auom-2014-0042

Keywords
Witness point set, Homotopy function, Gröbner basis, Local dimension, Monodromy action, Zero Sum Relation

© 2014 Shawki AL Rashed et al., published by De Gruyter Open

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Numerical Decomposition of Affine Algebraic Varieties

Published Online: Oct 20, 2015

Page range: 193 - 216

Received: Sep 01, 2012

Accepted: Feb 01, 2013

DOI: https://doi.org/10.2478/auom-2014-0042

KeywordsWitness point set, Homotopy function, Gröbner basis, Local dimension, Monodromy action, Zero Sum Relation

© 2014 Shawki AL Rashed et al., published by De Gruyter Open

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Keywords
Witness point set, Homotopy function, Gröbner basis, Local dimension, Monodromy action, Zero Sum Relation