On computing absolutely irreducible components of algebraic varieties with parameters

Abstract This paper presents a new algorithm for computing absolutely irreducible components of n-dimensional algebraic varieties defined implicitly by parametric homogeneous polynomial equations over $${\mathbb{Q}}$$, the field of rational numbers. The algorithm computes a finite partition of the p...
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Autor*in:	Ayad, Ali [verfasserIn]

Format:	Artikel
Sprache:	Englisch

Erschienen:	2010

Schlagwörter:	Symbolic computation Complexity analysis Parametric polynomials Algebraic varieties Irreducible components Effective generic points Polynomial factorization

Systematik:	SA 4220 SA 4220

Anmerkung:	© Springer-Verlag 2010

Übergeordnetes Werk:	Enthalten in: Computing - Springer Vienna, 1966, 89(2010), 1-2 vom: 22. Juni, Seite 45-68
Übergeordnetes Werk:	volume:89 ; year:2010 ; number:1-2 ; day:22 ; month:06 ; pages:45-68

Links:	Volltext

DOI / URN:	10.1007/s00607-010-0099-7

Katalog-ID:	OLC2061424945

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allfields	10.1007/s00607-010-0099-7 doi (DE-627)OLC2061424945 (DE-He213)s00607-010-0099-7-p DE-627 ger DE-627 rakwb eng 004 VZ SA 4220 VZ rvk SA 4220 VZ rvk Ayad, Ali verfasserin aut On computing absolutely irreducible components of algebraic varieties with parameters 2010 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer-Verlag 2010 Abstract This paper presents a new algorithm for computing absolutely irreducible components of n-dimensional algebraic varieties defined implicitly by parametric homogeneous polynomial equations over $${\mathbb{Q}}$$, the field of rational numbers. The algorithm computes a finite partition of the parameters space into constructible sets such that the absolutely irreducible components are given uniformly in each constructible set. Each component will be represented by two items: first by a parametric representative system, i.e., the equations that define the component and second by a parametric effective generic point which gives a parametric rational univariate representation of the elements of the component. The number of absolutely irreducible components is constant in each constructible set. The complexity bound of this algorithm is $${\delta^{O(r^4)}d^{r^4d^{O(n^3)}}}$$, being double exponential in n, where d (resp. δ) is an upper bound on the degrees of the input parametric polynomials w.r.t. the main n variables (resp. w.r.t. r parameters). Symbolic computation Complexity analysis Parametric polynomials Algebraic varieties Irreducible components Effective generic points Polynomial factorization Enthalten in Computing Springer Vienna, 1966 89(2010), 1-2 vom: 22. Juni, Seite 45-68 (DE-627)129534927 (DE-600)215907-7 (DE-576)014963949 0010-485X nnns volume:89 year:2010 number:1-2 day:22 month:06 pages:45-68 https://doi.org/10.1007/s00607-010-0099-7 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-TEC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_21 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_40 GBV_ILN_65 GBV_ILN_70 GBV_ILN_95 GBV_ILN_130 GBV_ILN_147 GBV_ILN_2004 GBV_ILN_2006 GBV_ILN_2012 GBV_ILN_2015 GBV_ILN_2057 GBV_ILN_2088 GBV_ILN_4036 GBV_ILN_4046 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4306 GBV_ILN_4311 GBV_ILN_4317 GBV_ILN_4318 GBV_ILN_4323 GBV_ILN_4325 GBV_ILN_4700 SA 4220 SA 4220 AR 89 2010 1-2 22 06 45-68
spelling	10.1007/s00607-010-0099-7 doi (DE-627)OLC2061424945 (DE-He213)s00607-010-0099-7-p DE-627 ger DE-627 rakwb eng 004 VZ SA 4220 VZ rvk SA 4220 VZ rvk Ayad, Ali verfasserin aut On computing absolutely irreducible components of algebraic varieties with parameters 2010 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer-Verlag 2010 Abstract This paper presents a new algorithm for computing absolutely irreducible components of n-dimensional algebraic varieties defined implicitly by parametric homogeneous polynomial equations over $${\mathbb{Q}}$$, the field of rational numbers. The algorithm computes a finite partition of the parameters space into constructible sets such that the absolutely irreducible components are given uniformly in each constructible set. Each component will be represented by two items: first by a parametric representative system, i.e., the equations that define the component and second by a parametric effective generic point which gives a parametric rational univariate representation of the elements of the component. The number of absolutely irreducible components is constant in each constructible set. The complexity bound of this algorithm is $${\delta^{O(r^4)}d^{r^4d^{O(n^3)}}}$$, being double exponential in n, where d (resp. δ) is an upper bound on the degrees of the input parametric polynomials w.r.t. the main n variables (resp. w.r.t. r parameters). Symbolic computation Complexity analysis Parametric polynomials Algebraic varieties Irreducible components Effective generic points Polynomial factorization Enthalten in Computing Springer Vienna, 1966 89(2010), 1-2 vom: 22. Juni, Seite 45-68 (DE-627)129534927 (DE-600)215907-7 (DE-576)014963949 0010-485X nnns volume:89 year:2010 number:1-2 day:22 month:06 pages:45-68 https://doi.org/10.1007/s00607-010-0099-7 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-TEC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_21 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_40 GBV_ILN_65 GBV_ILN_70 GBV_ILN_95 GBV_ILN_130 GBV_ILN_147 GBV_ILN_2004 GBV_ILN_2006 GBV_ILN_2012 GBV_ILN_2015 GBV_ILN_2057 GBV_ILN_2088 GBV_ILN_4036 GBV_ILN_4046 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4306 GBV_ILN_4311 GBV_ILN_4317 GBV_ILN_4318 GBV_ILN_4323 GBV_ILN_4325 GBV_ILN_4700 SA 4220 SA 4220 AR 89 2010 1-2 22 06 45-68
allfields_unstemmed	10.1007/s00607-010-0099-7 doi (DE-627)OLC2061424945 (DE-He213)s00607-010-0099-7-p DE-627 ger DE-627 rakwb eng 004 VZ SA 4220 VZ rvk SA 4220 VZ rvk Ayad, Ali verfasserin aut On computing absolutely irreducible components of algebraic varieties with parameters 2010 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer-Verlag 2010 Abstract This paper presents a new algorithm for computing absolutely irreducible components of n-dimensional algebraic varieties defined implicitly by parametric homogeneous polynomial equations over $${\mathbb{Q}}$$, the field of rational numbers. The algorithm computes a finite partition of the parameters space into constructible sets such that the absolutely irreducible components are given uniformly in each constructible set. Each component will be represented by two items: first by a parametric representative system, i.e., the equations that define the component and second by a parametric effective generic point which gives a parametric rational univariate representation of the elements of the component. The number of absolutely irreducible components is constant in each constructible set. The complexity bound of this algorithm is $${\delta^{O(r^4)}d^{r^4d^{O(n^3)}}}$$, being double exponential in n, where d (resp. δ) is an upper bound on the degrees of the input parametric polynomials w.r.t. the main n variables (resp. w.r.t. r parameters). Symbolic computation Complexity analysis Parametric polynomials Algebraic varieties Irreducible components Effective generic points Polynomial factorization Enthalten in Computing Springer Vienna, 1966 89(2010), 1-2 vom: 22. Juni, Seite 45-68 (DE-627)129534927 (DE-600)215907-7 (DE-576)014963949 0010-485X nnns volume:89 year:2010 number:1-2 day:22 month:06 pages:45-68 https://doi.org/10.1007/s00607-010-0099-7 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-TEC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_21 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_40 GBV_ILN_65 GBV_ILN_70 GBV_ILN_95 GBV_ILN_130 GBV_ILN_147 GBV_ILN_2004 GBV_ILN_2006 GBV_ILN_2012 GBV_ILN_2015 GBV_ILN_2057 GBV_ILN_2088 GBV_ILN_4036 GBV_ILN_4046 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4306 GBV_ILN_4311 GBV_ILN_4317 GBV_ILN_4318 GBV_ILN_4323 GBV_ILN_4325 GBV_ILN_4700 SA 4220 SA 4220 AR 89 2010 1-2 22 06 45-68
allfieldsGer	10.1007/s00607-010-0099-7 doi (DE-627)OLC2061424945 (DE-He213)s00607-010-0099-7-p DE-627 ger DE-627 rakwb eng 004 VZ SA 4220 VZ rvk SA 4220 VZ rvk Ayad, Ali verfasserin aut On computing absolutely irreducible components of algebraic varieties with parameters 2010 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer-Verlag 2010 Abstract This paper presents a new algorithm for computing absolutely irreducible components of n-dimensional algebraic varieties defined implicitly by parametric homogeneous polynomial equations over $${\mathbb{Q}}$$, the field of rational numbers. The algorithm computes a finite partition of the parameters space into constructible sets such that the absolutely irreducible components are given uniformly in each constructible set. Each component will be represented by two items: first by a parametric representative system, i.e., the equations that define the component and second by a parametric effective generic point which gives a parametric rational univariate representation of the elements of the component. The number of absolutely irreducible components is constant in each constructible set. The complexity bound of this algorithm is $${\delta^{O(r^4)}d^{r^4d^{O(n^3)}}}$$, being double exponential in n, where d (resp. δ) is an upper bound on the degrees of the input parametric polynomials w.r.t. the main n variables (resp. w.r.t. r parameters). Symbolic computation Complexity analysis Parametric polynomials Algebraic varieties Irreducible components Effective generic points Polynomial factorization Enthalten in Computing Springer Vienna, 1966 89(2010), 1-2 vom: 22. Juni, Seite 45-68 (DE-627)129534927 (DE-600)215907-7 (DE-576)014963949 0010-485X nnns volume:89 year:2010 number:1-2 day:22 month:06 pages:45-68 https://doi.org/10.1007/s00607-010-0099-7 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-TEC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_21 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_40 GBV_ILN_65 GBV_ILN_70 GBV_ILN_95 GBV_ILN_130 GBV_ILN_147 GBV_ILN_2004 GBV_ILN_2006 GBV_ILN_2012 GBV_ILN_2015 GBV_ILN_2057 GBV_ILN_2088 GBV_ILN_4036 GBV_ILN_4046 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4306 GBV_ILN_4311 GBV_ILN_4317 GBV_ILN_4318 GBV_ILN_4323 GBV_ILN_4325 GBV_ILN_4700 SA 4220 SA 4220 AR 89 2010 1-2 22 06 45-68
allfieldsSound	10.1007/s00607-010-0099-7 doi (DE-627)OLC2061424945 (DE-He213)s00607-010-0099-7-p DE-627 ger DE-627 rakwb eng 004 VZ SA 4220 VZ rvk SA 4220 VZ rvk Ayad, Ali verfasserin aut On computing absolutely irreducible components of algebraic varieties with parameters 2010 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer-Verlag 2010 Abstract This paper presents a new algorithm for computing absolutely irreducible components of n-dimensional algebraic varieties defined implicitly by parametric homogeneous polynomial equations over $${\mathbb{Q}}$$, the field of rational numbers. The algorithm computes a finite partition of the parameters space into constructible sets such that the absolutely irreducible components are given uniformly in each constructible set. Each component will be represented by two items: first by a parametric representative system, i.e., the equations that define the component and second by a parametric effective generic point which gives a parametric rational univariate representation of the elements of the component. The number of absolutely irreducible components is constant in each constructible set. The complexity bound of this algorithm is $${\delta^{O(r^4)}d^{r^4d^{O(n^3)}}}$$, being double exponential in n, where d (resp. δ) is an upper bound on the degrees of the input parametric polynomials w.r.t. the main n variables (resp. w.r.t. r parameters). Symbolic computation Complexity analysis Parametric polynomials Algebraic varieties Irreducible components Effective generic points Polynomial factorization Enthalten in Computing Springer Vienna, 1966 89(2010), 1-2 vom: 22. Juni, Seite 45-68 (DE-627)129534927 (DE-600)215907-7 (DE-576)014963949 0010-485X nnns volume:89 year:2010 number:1-2 day:22 month:06 pages:45-68 https://doi.org/10.1007/s00607-010-0099-7 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-TEC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_21 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_40 GBV_ILN_65 GBV_ILN_70 GBV_ILN_95 GBV_ILN_130 GBV_ILN_147 GBV_ILN_2004 GBV_ILN_2006 GBV_ILN_2012 GBV_ILN_2015 GBV_ILN_2057 GBV_ILN_2088 GBV_ILN_4036 GBV_ILN_4046 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4306 GBV_ILN_4311 GBV_ILN_4317 GBV_ILN_4318 GBV_ILN_4323 GBV_ILN_4325 GBV_ILN_4700 SA 4220 SA 4220 AR 89 2010 1-2 22 06 45-68
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source	Enthalten in Computing 89(2010), 1-2 vom: 22. Juni, Seite 45-68 volume:89 year:2010 number:1-2 day:22 month:06 pages:45-68
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topic_facet	Symbolic computation Complexity analysis Parametric polynomials Algebraic varieties Irreducible components Effective generic points Polynomial factorization
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author	Ayad, Ali
spellingShingle	Ayad, Ali ddc 004 rvk SA 4220 misc Symbolic computation misc Complexity analysis misc Parametric polynomials misc Algebraic varieties misc Irreducible components misc Effective generic points misc Polynomial factorization On computing absolutely irreducible components of algebraic varieties with parameters
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topic_title	004 VZ SA 4220 VZ rvk On computing absolutely irreducible components of algebraic varieties with parameters Symbolic computation Complexity analysis Parametric polynomials Algebraic varieties Irreducible components Effective generic points Polynomial factorization
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topic_unstemmed	ddc 004 rvk SA 4220 misc Symbolic computation misc Complexity analysis misc Parametric polynomials misc Algebraic varieties misc Irreducible components misc Effective generic points misc Polynomial factorization
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title	On computing absolutely irreducible components of algebraic varieties with parameters
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title_full	On computing absolutely irreducible components of algebraic varieties with parameters
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title_sort	on computing absolutely irreducible components of algebraic varieties with parameters
title_auth	On computing absolutely irreducible components of algebraic varieties with parameters
abstract	Abstract This paper presents a new algorithm for computing absolutely irreducible components of n-dimensional algebraic varieties defined implicitly by parametric homogeneous polynomial equations over $${\mathbb{Q}}$$, the field of rational numbers. The algorithm computes a finite partition of the parameters space into constructible sets such that the absolutely irreducible components are given uniformly in each constructible set. Each component will be represented by two items: first by a parametric representative system, i.e., the equations that define the component and second by a parametric effective generic point which gives a parametric rational univariate representation of the elements of the component. The number of absolutely irreducible components is constant in each constructible set. The complexity bound of this algorithm is $${\delta^{O(r^4)}d^{r^4d^{O(n^3)}}}$$, being double exponential in n, where d (resp. δ) is an upper bound on the degrees of the input parametric polynomials w.r.t. the main n variables (resp. w.r.t. r parameters). © Springer-Verlag 2010
abstractGer	Abstract This paper presents a new algorithm for computing absolutely irreducible components of n-dimensional algebraic varieties defined implicitly by parametric homogeneous polynomial equations over $${\mathbb{Q}}$$, the field of rational numbers. The algorithm computes a finite partition of the parameters space into constructible sets such that the absolutely irreducible components are given uniformly in each constructible set. Each component will be represented by two items: first by a parametric representative system, i.e., the equations that define the component and second by a parametric effective generic point which gives a parametric rational univariate representation of the elements of the component. The number of absolutely irreducible components is constant in each constructible set. The complexity bound of this algorithm is $${\delta^{O(r^4)}d^{r^4d^{O(n^3)}}}$$, being double exponential in n, where d (resp. δ) is an upper bound on the degrees of the input parametric polynomials w.r.t. the main n variables (resp. w.r.t. r parameters). © Springer-Verlag 2010
abstract_unstemmed	Abstract This paper presents a new algorithm for computing absolutely irreducible components of n-dimensional algebraic varieties defined implicitly by parametric homogeneous polynomial equations over $${\mathbb{Q}}$$, the field of rational numbers. The algorithm computes a finite partition of the parameters space into constructible sets such that the absolutely irreducible components are given uniformly in each constructible set. Each component will be represented by two items: first by a parametric representative system, i.e., the equations that define the component and second by a parametric effective generic point which gives a parametric rational univariate representation of the elements of the component. The number of absolutely irreducible components is constant in each constructible set. The complexity bound of this algorithm is $${\delta^{O(r^4)}d^{r^4d^{O(n^3)}}}$$, being double exponential in n, where d (resp. δ) is an upper bound on the degrees of the input parametric polynomials w.r.t. the main n variables (resp. w.r.t. r parameters). © Springer-Verlag 2010
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title_short	On computing absolutely irreducible components of algebraic varieties with parameters
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