A dimension series for multivariate splines

Abstract For a polyhedral subdivision Δ of a region in Euclideand-space, we consider the vector spaceCkr(Δ) consisting of allCr piecewise polynomial functions over Δ of degree at mostk. We consider the formal power series ∑k≥0 $ dim_{ℝ} $ Ckr(Δ)$ λ^{k} $ and show, under mild conditions on Δ, that th...
Ausführliche Beschreibung

Gespeichert in:

Autor*in:	Billera, Louis J. [verfasserIn] Rose, Lauren L. [verfasserIn]

Format:	E-Artikel
Sprache:	Englisch

Erschienen:	1991

Schlagwörter:	Polynomial Ring Formal Power Series Hilbert Series Noetherian Ring Hilbert Function

Übergeordnetes Werk:	Enthalten in: Discrete & computational geometry - New York, NY : Springer, 1986, 6(1991), 2 vom: Juni, Seite 107-128
Übergeordnetes Werk:	volume:6 ; year:1991 ; number:2 ; month:06 ; pages:107-128

Links:	Volltext

DOI / URN:	10.1007/BF02574678

Katalog-ID:	SPR006185827

Internformat


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100	1		\|a Billera, Louis J. \|e verfasserin \|4 aut
245	1	2	\|a A dimension series for multivariate splines
264		1	\|c 1991
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520			\|a Abstract For a polyhedral subdivision Δ of a region in Euclideand-space, we consider the vector spaceCkr(Δ) consisting of allCr piecewise polynomial functions over Δ of degree at mostk. We consider the formal power series ∑k≥0 $ dim_{ℝ} $ Ckr(Δ)$ λ^{k} $ and show, under mild conditions on Δ, that this always has the formP(λ)/(1−λ)d+1, whereP(λ) is a polynomial in λ with integral coefficients which satisfiesP(0)=1,P(1)=fd (Δ), andP′(1)=(r+1)fd−10(Δ). We discuss how the polynomialP(λ) and bases for the spacesCkr(Δ) can be effectively calculated by use of Gröbner basis techniques of computational commutative algebra. A further application is given to the theory of hyperplane arrangements.
650		4	\|a Polynomial Ring \|7 (dpeaa)DE-He213
650		4	\|a Formal Power Series \|7 (dpeaa)DE-He213
650		4	\|a Hilbert Series \|7 (dpeaa)DE-He213
650		4	\|a Noetherian Ring \|7 (dpeaa)DE-He213
650		4	\|a Hilbert Function \|7 (dpeaa)DE-He213
700	1		\|a Rose, Lauren L. \|e verfasserin \|4 aut
773	0	8	\|i Enthalten in \|t Discrete & computational geometry \|d New York, NY : Springer, 1986 \|g 6(1991), 2 vom: Juni, Seite 107-128 \|w (DE-627)253722330 \|w (DE-600)1459007-4 \|x 1432-0444 \|7 nnns
773	1	8	\|g volume:6 \|g year:1991 \|g number:2 \|g month:06 \|g pages:107-128
856	4	0	\|u https://dx.doi.org/10.1007/BF02574678 \|z lizenzpflichtig \|3 Volltext
912			\|a GBV_USEFLAG_A
912			\|a SYSFLAG_A
912			\|a GBV_SPRINGER
912			\|a SSG-OPC-MAT
912			\|a SSG-OPC-ASE
912			\|a GBV_ILN_11
912			\|a GBV_ILN_20
912			\|a GBV_ILN_22
912			\|a GBV_ILN_23
912			\|a GBV_ILN_24
912			\|a GBV_ILN_31
912			\|a GBV_ILN_32
912			\|a GBV_ILN_39
912			\|a GBV_ILN_40
912			\|a GBV_ILN_60
912			\|a GBV_ILN_62
912			\|a GBV_ILN_63
912			\|a GBV_ILN_69
912			\|a GBV_ILN_70
912			\|a GBV_ILN_73
912			\|a GBV_ILN_74
912			\|a GBV_ILN_90
912			\|a GBV_ILN_95
912			\|a GBV_ILN_100
912			\|a GBV_ILN_101
912			\|a GBV_ILN_105
912			\|a GBV_ILN_110
912			\|a GBV_ILN_120
912			\|a GBV_ILN_121
912			\|a GBV_ILN_138
912			\|a GBV_ILN_150
912			\|a GBV_ILN_151
912			\|a GBV_ILN_152
912			\|a GBV_ILN_161
912			\|a GBV_ILN_170
912			\|a GBV_ILN_171
912			\|a GBV_ILN_187
912			\|a GBV_ILN_206
912			\|a GBV_ILN_224
912			\|a GBV_ILN_285
912			\|a GBV_ILN_293
912			\|a GBV_ILN_370
912			\|a GBV_ILN_374
912			\|a GBV_ILN_602
912			\|a GBV_ILN_647
912			\|a GBV_ILN_702
912			\|a GBV_ILN_2001
912			\|a GBV_ILN_2003
912			\|a GBV_ILN_2004
912			\|a GBV_ILN_2005
912			\|a GBV_ILN_2006
912			\|a GBV_ILN_2007
912			\|a GBV_ILN_2008
912			\|a GBV_ILN_2009
912			\|a GBV_ILN_2010
912			\|a GBV_ILN_2011
912			\|a GBV_ILN_2014
912			\|a GBV_ILN_2015
912			\|a GBV_ILN_2018
912			\|a GBV_ILN_2020
912			\|a GBV_ILN_2021
912			\|a GBV_ILN_2025
912			\|a GBV_ILN_2026
912			\|a GBV_ILN_2027
912			\|a GBV_ILN_2031
912			\|a GBV_ILN_2034
912			\|a GBV_ILN_2037
912			\|a GBV_ILN_2038
912			\|a GBV_ILN_2039
912			\|a GBV_ILN_2043
912			\|a GBV_ILN_2044
912			\|a GBV_ILN_2048
912			\|a GBV_ILN_2050
912			\|a GBV_ILN_2055
912			\|a GBV_ILN_2056
912			\|a GBV_ILN_2057
912			\|a GBV_ILN_2059
912			\|a GBV_ILN_2061
912			\|a GBV_ILN_2064
912			\|a GBV_ILN_2065
912			\|a GBV_ILN_2068
912			\|a GBV_ILN_2088
912			\|a GBV_ILN_2093
912			\|a GBV_ILN_2106
912			\|a GBV_ILN_2107
912			\|a GBV_ILN_2108
912			\|a GBV_ILN_2110
912			\|a GBV_ILN_2111
912			\|a GBV_ILN_2112
912			\|a GBV_ILN_2113
912			\|a GBV_ILN_2116
912			\|a GBV_ILN_2118
912			\|a GBV_ILN_2119
912			\|a GBV_ILN_2122
912			\|a GBV_ILN_2129
912			\|a GBV_ILN_2143
912			\|a GBV_ILN_2144
912			\|a GBV_ILN_2147
912			\|a GBV_ILN_2148
912			\|a GBV_ILN_2152
912			\|a GBV_ILN_2153
912			\|a GBV_ILN_2158
912			\|a GBV_ILN_2188
912			\|a GBV_ILN_2190
912			\|a GBV_ILN_2193
912			\|a GBV_ILN_2232
912			\|a GBV_ILN_2336
912			\|a GBV_ILN_2446
912			\|a GBV_ILN_2470
912			\|a GBV_ILN_2472
912			\|a GBV_ILN_2507
912			\|a GBV_ILN_2522
912			\|a GBV_ILN_2548
912			\|a GBV_ILN_2808
912			\|a GBV_ILN_4012
912			\|a GBV_ILN_4035
912			\|a GBV_ILN_4037
912			\|a GBV_ILN_4046
912			\|a GBV_ILN_4112
912			\|a GBV_ILN_4125
912			\|a GBV_ILN_4126
912			\|a GBV_ILN_4242
912			\|a GBV_ILN_4246
912			\|a GBV_ILN_4249
912			\|a GBV_ILN_4251
912			\|a GBV_ILN_4277
912			\|a GBV_ILN_4305
912			\|a GBV_ILN_4306
912			\|a GBV_ILN_4307
912			\|a GBV_ILN_4313
912			\|a GBV_ILN_4322
912			\|a GBV_ILN_4323
912			\|a GBV_ILN_4324
912			\|a GBV_ILN_4325
912			\|a GBV_ILN_4326
912			\|a GBV_ILN_4328
912			\|a GBV_ILN_4333
912			\|a GBV_ILN_4334
912			\|a GBV_ILN_4335
912			\|a GBV_ILN_4336
912			\|a GBV_ILN_4338
912			\|a GBV_ILN_4346
912			\|a GBV_ILN_4367
912			\|a GBV_ILN_4393
912			\|a GBV_ILN_4700
912			\|a GBV_ILN_4753
936	b	k	\|a 31.59 \|q ASE
936	b	k	\|a 31.12 \|q ASE
936	b	k	\|a 31.76 \|q ASE
951			\|a AR
952			\|d 6 \|j 1991 \|e 2 \|c 06 \|h 107-128

Indexfelder

author_variant	l j b lj ljb l l r ll llr
matchkey_str	article:14320444:1991----::dmnineisomlia
hierarchy_sort_str	1991
bklnumber	31.59 31.12 31.76
publishDate	1991
allfields	10.1007/BF02574678 doi (DE-627)SPR006185827 (SPR)BF02574678-e DE-627 ger DE-627 rakwb eng 510 004 ASE 510 004 ASE 31.59 bkl 31.12 bkl 31.76 bkl Billera, Louis J. verfasserin aut A dimension series for multivariate splines 1991 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier Abstract For a polyhedral subdivision Δ of a region in Euclideand-space, we consider the vector spaceCkr(Δ) consisting of allCr piecewise polynomial functions over Δ of degree at mostk. We consider the formal power series ∑k≥0 $ dim_{ℝ} $ Ckr(Δ)$ λ^{k} $ and show, under mild conditions on Δ, that this always has the formP(λ)/(1−λ)d+1, whereP(λ) is a polynomial in λ with integral coefficients which satisfiesP(0)=1,P(1)=fd (Δ), andP′(1)=(r+1)fd−10(Δ). We discuss how the polynomialP(λ) and bases for the spacesCkr(Δ) can be effectively calculated by use of Gröbner basis techniques of computational commutative algebra. A further application is given to the theory of hyperplane arrangements. Polynomial Ring (dpeaa)DE-He213 Formal Power Series (dpeaa)DE-He213 Hilbert Series (dpeaa)DE-He213 Noetherian Ring (dpeaa)DE-He213 Hilbert Function (dpeaa)DE-He213 Rose, Lauren L. verfasserin aut Enthalten in Discrete & computational geometry New York, NY : Springer, 1986 6(1991), 2 vom: Juni, Seite 107-128 (DE-627)253722330 (DE-600)1459007-4 1432-0444 nnns volume:6 year:1991 number:2 month:06 pages:107-128 https://dx.doi.org/10.1007/BF02574678 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER SSG-OPC-MAT SSG-OPC-ASE GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_101 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_121 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_206 GBV_ILN_224 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_374 GBV_ILN_602 GBV_ILN_647 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2018 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2043 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2116 GBV_ILN_2118 GBV_ILN_2119 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2158 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2193 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_2808 GBV_ILN_4012 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4277 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4346 GBV_ILN_4367 GBV_ILN_4393 GBV_ILN_4700 GBV_ILN_4753 31.59 ASE 31.12 ASE 31.76 ASE AR 6 1991 2 06 107-128
spelling	10.1007/BF02574678 doi (DE-627)SPR006185827 (SPR)BF02574678-e DE-627 ger DE-627 rakwb eng 510 004 ASE 510 004 ASE 31.59 bkl 31.12 bkl 31.76 bkl Billera, Louis J. verfasserin aut A dimension series for multivariate splines 1991 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier Abstract For a polyhedral subdivision Δ of a region in Euclideand-space, we consider the vector spaceCkr(Δ) consisting of allCr piecewise polynomial functions over Δ of degree at mostk. We consider the formal power series ∑k≥0 $ dim_{ℝ} $ Ckr(Δ)$ λ^{k} $ and show, under mild conditions on Δ, that this always has the formP(λ)/(1−λ)d+1, whereP(λ) is a polynomial in λ with integral coefficients which satisfiesP(0)=1,P(1)=fd (Δ), andP′(1)=(r+1)fd−10(Δ). We discuss how the polynomialP(λ) and bases for the spacesCkr(Δ) can be effectively calculated by use of Gröbner basis techniques of computational commutative algebra. A further application is given to the theory of hyperplane arrangements. Polynomial Ring (dpeaa)DE-He213 Formal Power Series (dpeaa)DE-He213 Hilbert Series (dpeaa)DE-He213 Noetherian Ring (dpeaa)DE-He213 Hilbert Function (dpeaa)DE-He213 Rose, Lauren L. verfasserin aut Enthalten in Discrete & computational geometry New York, NY : Springer, 1986 6(1991), 2 vom: Juni, Seite 107-128 (DE-627)253722330 (DE-600)1459007-4 1432-0444 nnns volume:6 year:1991 number:2 month:06 pages:107-128 https://dx.doi.org/10.1007/BF02574678 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER SSG-OPC-MAT SSG-OPC-ASE GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_101 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_121 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_206 GBV_ILN_224 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_374 GBV_ILN_602 GBV_ILN_647 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2018 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2043 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2116 GBV_ILN_2118 GBV_ILN_2119 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2158 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2193 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_2808 GBV_ILN_4012 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4277 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4346 GBV_ILN_4367 GBV_ILN_4393 GBV_ILN_4700 GBV_ILN_4753 31.59 ASE 31.12 ASE 31.76 ASE AR 6 1991 2 06 107-128
allfields_unstemmed	10.1007/BF02574678 doi (DE-627)SPR006185827 (SPR)BF02574678-e DE-627 ger DE-627 rakwb eng 510 004 ASE 510 004 ASE 31.59 bkl 31.12 bkl 31.76 bkl Billera, Louis J. verfasserin aut A dimension series for multivariate splines 1991 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier Abstract For a polyhedral subdivision Δ of a region in Euclideand-space, we consider the vector spaceCkr(Δ) consisting of allCr piecewise polynomial functions over Δ of degree at mostk. We consider the formal power series ∑k≥0 $ dim_{ℝ} $ Ckr(Δ)$ λ^{k} $ and show, under mild conditions on Δ, that this always has the formP(λ)/(1−λ)d+1, whereP(λ) is a polynomial in λ with integral coefficients which satisfiesP(0)=1,P(1)=fd (Δ), andP′(1)=(r+1)fd−10(Δ). We discuss how the polynomialP(λ) and bases for the spacesCkr(Δ) can be effectively calculated by use of Gröbner basis techniques of computational commutative algebra. A further application is given to the theory of hyperplane arrangements. Polynomial Ring (dpeaa)DE-He213 Formal Power Series (dpeaa)DE-He213 Hilbert Series (dpeaa)DE-He213 Noetherian Ring (dpeaa)DE-He213 Hilbert Function (dpeaa)DE-He213 Rose, Lauren L. verfasserin aut Enthalten in Discrete & computational geometry New York, NY : Springer, 1986 6(1991), 2 vom: Juni, Seite 107-128 (DE-627)253722330 (DE-600)1459007-4 1432-0444 nnns volume:6 year:1991 number:2 month:06 pages:107-128 https://dx.doi.org/10.1007/BF02574678 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER SSG-OPC-MAT SSG-OPC-ASE GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_101 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_121 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_206 GBV_ILN_224 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_374 GBV_ILN_602 GBV_ILN_647 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2018 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2043 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2116 GBV_ILN_2118 GBV_ILN_2119 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2158 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2193 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_2808 GBV_ILN_4012 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4277 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4346 GBV_ILN_4367 GBV_ILN_4393 GBV_ILN_4700 GBV_ILN_4753 31.59 ASE 31.12 ASE 31.76 ASE AR 6 1991 2 06 107-128
allfieldsGer	10.1007/BF02574678 doi (DE-627)SPR006185827 (SPR)BF02574678-e DE-627 ger DE-627 rakwb eng 510 004 ASE 510 004 ASE 31.59 bkl 31.12 bkl 31.76 bkl Billera, Louis J. verfasserin aut A dimension series for multivariate splines 1991 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier Abstract For a polyhedral subdivision Δ of a region in Euclideand-space, we consider the vector spaceCkr(Δ) consisting of allCr piecewise polynomial functions over Δ of degree at mostk. We consider the formal power series ∑k≥0 $ dim_{ℝ} $ Ckr(Δ)$ λ^{k} $ and show, under mild conditions on Δ, that this always has the formP(λ)/(1−λ)d+1, whereP(λ) is a polynomial in λ with integral coefficients which satisfiesP(0)=1,P(1)=fd (Δ), andP′(1)=(r+1)fd−10(Δ). We discuss how the polynomialP(λ) and bases for the spacesCkr(Δ) can be effectively calculated by use of Gröbner basis techniques of computational commutative algebra. A further application is given to the theory of hyperplane arrangements. Polynomial Ring (dpeaa)DE-He213 Formal Power Series (dpeaa)DE-He213 Hilbert Series (dpeaa)DE-He213 Noetherian Ring (dpeaa)DE-He213 Hilbert Function (dpeaa)DE-He213 Rose, Lauren L. verfasserin aut Enthalten in Discrete & computational geometry New York, NY : Springer, 1986 6(1991), 2 vom: Juni, Seite 107-128 (DE-627)253722330 (DE-600)1459007-4 1432-0444 nnns volume:6 year:1991 number:2 month:06 pages:107-128 https://dx.doi.org/10.1007/BF02574678 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER SSG-OPC-MAT SSG-OPC-ASE GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_101 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_121 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_206 GBV_ILN_224 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_374 GBV_ILN_602 GBV_ILN_647 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2018 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2043 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2116 GBV_ILN_2118 GBV_ILN_2119 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2158 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2193 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_2808 GBV_ILN_4012 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4277 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4346 GBV_ILN_4367 GBV_ILN_4393 GBV_ILN_4700 GBV_ILN_4753 31.59 ASE 31.12 ASE 31.76 ASE AR 6 1991 2 06 107-128
allfieldsSound	10.1007/BF02574678 doi (DE-627)SPR006185827 (SPR)BF02574678-e DE-627 ger DE-627 rakwb eng 510 004 ASE 510 004 ASE 31.59 bkl 31.12 bkl 31.76 bkl Billera, Louis J. verfasserin aut A dimension series for multivariate splines 1991 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier Abstract For a polyhedral subdivision Δ of a region in Euclideand-space, we consider the vector spaceCkr(Δ) consisting of allCr piecewise polynomial functions over Δ of degree at mostk. We consider the formal power series ∑k≥0 $ dim_{ℝ} $ Ckr(Δ)$ λ^{k} $ and show, under mild conditions on Δ, that this always has the formP(λ)/(1−λ)d+1, whereP(λ) is a polynomial in λ with integral coefficients which satisfiesP(0)=1,P(1)=fd (Δ), andP′(1)=(r+1)fd−10(Δ). We discuss how the polynomialP(λ) and bases for the spacesCkr(Δ) can be effectively calculated by use of Gröbner basis techniques of computational commutative algebra. A further application is given to the theory of hyperplane arrangements. Polynomial Ring (dpeaa)DE-He213 Formal Power Series (dpeaa)DE-He213 Hilbert Series (dpeaa)DE-He213 Noetherian Ring (dpeaa)DE-He213 Hilbert Function (dpeaa)DE-He213 Rose, Lauren L. verfasserin aut Enthalten in Discrete & computational geometry New York, NY : Springer, 1986 6(1991), 2 vom: Juni, Seite 107-128 (DE-627)253722330 (DE-600)1459007-4 1432-0444 nnns volume:6 year:1991 number:2 month:06 pages:107-128 https://dx.doi.org/10.1007/BF02574678 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER SSG-OPC-MAT SSG-OPC-ASE GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_101 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_121 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_206 GBV_ILN_224 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_374 GBV_ILN_602 GBV_ILN_647 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2018 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2043 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2116 GBV_ILN_2118 GBV_ILN_2119 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2158 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2193 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_2808 GBV_ILN_4012 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4277 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4346 GBV_ILN_4367 GBV_ILN_4393 GBV_ILN_4700 GBV_ILN_4753 31.59 ASE 31.12 ASE 31.76 ASE AR 6 1991 2 06 107-128
language	English
source	Enthalten in Discrete & computational geometry 6(1991), 2 vom: Juni, Seite 107-128 volume:6 year:1991 number:2 month:06 pages:107-128
sourceStr	Enthalten in Discrete & computational geometry 6(1991), 2 vom: Juni, Seite 107-128 volume:6 year:1991 number:2 month:06 pages:107-128
format_phy_str_mv	Article
institution	findex.gbv.de
topic_facet	Polynomial Ring Formal Power Series Hilbert Series Noetherian Ring Hilbert Function
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container_title	Discrete & computational geometry
authorswithroles_txt_mv	Billera, Louis J. @@aut@@ Rose, Lauren L. @@aut@@
publishDateDaySort_date	1991-06-01T00:00:00Z
hierarchy_top_id	253722330
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author	Billera, Louis J.
spellingShingle	Billera, Louis J. ddc 510 bkl 31.59 bkl 31.12 bkl 31.76 misc Polynomial Ring misc Formal Power Series misc Hilbert Series misc Noetherian Ring misc Hilbert Function A dimension series for multivariate splines
authorStr	Billera, Louis J.
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topic_title	510 004 ASE 31.59 bkl 31.12 bkl 31.76 bkl A dimension series for multivariate splines Polynomial Ring (dpeaa)DE-He213 Formal Power Series (dpeaa)DE-He213 Hilbert Series (dpeaa)DE-He213 Noetherian Ring (dpeaa)DE-He213 Hilbert Function (dpeaa)DE-He213
topic	ddc 510 bkl 31.59 bkl 31.12 bkl 31.76 misc Polynomial Ring misc Formal Power Series misc Hilbert Series misc Noetherian Ring misc Hilbert Function
topic_unstemmed	ddc 510 bkl 31.59 bkl 31.12 bkl 31.76 misc Polynomial Ring misc Formal Power Series misc Hilbert Series misc Noetherian Ring misc Hilbert Function
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title	A dimension series for multivariate splines
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title_full	A dimension series for multivariate splines
author_sort	Billera, Louis J.
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author-letter	Billera, Louis J.
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dewey-full	510 004
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title_sort	dimension series for multivariate splines
title_auth	A dimension series for multivariate splines
abstract	Abstract For a polyhedral subdivision Δ of a region in Euclideand-space, we consider the vector spaceCkr(Δ) consisting of allCr piecewise polynomial functions over Δ of degree at mostk. We consider the formal power series ∑k≥0 $ dim_{ℝ} $ Ckr(Δ)$ λ^{k} $ and show, under mild conditions on Δ, that this always has the formP(λ)/(1−λ)d+1, whereP(λ) is a polynomial in λ with integral coefficients which satisfiesP(0)=1,P(1)=fd (Δ), andP′(1)=(r+1)fd−10(Δ). We discuss how the polynomialP(λ) and bases for the spacesCkr(Δ) can be effectively calculated by use of Gröbner basis techniques of computational commutative algebra. A further application is given to the theory of hyperplane arrangements.
abstractGer	Abstract For a polyhedral subdivision Δ of a region in Euclideand-space, we consider the vector spaceCkr(Δ) consisting of allCr piecewise polynomial functions over Δ of degree at mostk. We consider the formal power series ∑k≥0 $ dim_{ℝ} $ Ckr(Δ)$ λ^{k} $ and show, under mild conditions on Δ, that this always has the formP(λ)/(1−λ)d+1, whereP(λ) is a polynomial in λ with integral coefficients which satisfiesP(0)=1,P(1)=fd (Δ), andP′(1)=(r+1)fd−10(Δ). We discuss how the polynomialP(λ) and bases for the spacesCkr(Δ) can be effectively calculated by use of Gröbner basis techniques of computational commutative algebra. A further application is given to the theory of hyperplane arrangements.
abstract_unstemmed	Abstract For a polyhedral subdivision Δ of a region in Euclideand-space, we consider the vector spaceCkr(Δ) consisting of allCr piecewise polynomial functions over Δ of degree at mostk. We consider the formal power series ∑k≥0 $ dim_{ℝ} $ Ckr(Δ)$ λ^{k} $ and show, under mild conditions on Δ, that this always has the formP(λ)/(1−λ)d+1, whereP(λ) is a polynomial in λ with integral coefficients which satisfiesP(0)=1,P(1)=fd (Δ), andP′(1)=(r+1)fd−10(Δ). We discuss how the polynomialP(λ) and bases for the spacesCkr(Δ) can be effectively calculated by use of Gröbner basis techniques of computational commutative algebra. A further application is given to the theory of hyperplane arrangements.
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container_issue	2
title_short	A dimension series for multivariate splines
url	https://dx.doi.org/10.1007/BF02574678
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author2	Rose, Lauren L.
author2Str	Rose, Lauren L.
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doi_str	10.1007/BF02574678
up_date	2024-07-03T21:24:32.770Z
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A dimension series for multivariate splines

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