Sum of squares length of real forms

Abstract For %$n,\,d\ge 1%$ let p(n, 2d) denote the smallest number p such that every sum of squares of degree d forms in %${\mathbb {R}}[x_1,\ldots ,x_n]%$ is a sum of p squares. We establish lower bounds for p(n, 2d) that are considerably stronger than the bounds known so far. Combined with known...
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Gespeichert in:

Autor*in:	Scheiderer, Claus [verfasserIn]

Format:	E-Artikel
Sprache:	Englisch

Erschienen:	2016

Schlagwörter:	Lower Bound Homogeneous Polynomial Hilbert Series Hilbert Function Artinian Ring

Anmerkung:	© Springer-Verlag Berlin Heidelberg 2016

Übergeordnetes Werk:	Enthalten in: Mathematische Zeitschrift - Berlin : Springer, 1918, 286(2016), 1-2 vom: 27. Okt., Seite 559-570
Übergeordnetes Werk:	volume:286 ; year:2016 ; number:1-2 ; day:27 ; month:10 ; pages:559-570

Links:	Volltext

DOI / URN:	10.1007/s00209-016-1773-z

Katalog-ID:	SPR001930575

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520			\|a Abstract For %$n,\,d\ge 1%$ let p(n, 2d) denote the smallest number p such that every sum of squares of degree d forms in %${\mathbb {R}}[x_1,\ldots ,x_n]%$ is a sum of p squares. We establish lower bounds for p(n, 2d) that are considerably stronger than the bounds known so far. Combined with known upper bounds they give %$p(3,2d)\in \{d+1,\,d+2\}%$ in the ternary case. Assuming a conjecture of Iarrobino–Kanev on dimensions of tangent spaces to catalecticant varieties, we show that %$p(n,2d)\sim const\cdot d^{(n-1)/2}%$ for %$d\rightarrow \infty %$ and all %$n\ge 3%$. For ternary sextics and quaternary quartics we determine the exact value of the invariant, showing %$p(3,6)=4%$ and %$p(4,4)=5%$.
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650		4	\|a Artinian Ring \|7 (dpeaa)DE-He213
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912			\|a GBV_ILN_161
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912			\|a GBV_ILN_224
912			\|a GBV_ILN_230
912			\|a GBV_ILN_250
912			\|a GBV_ILN_267
912			\|a GBV_ILN_281
912			\|a GBV_ILN_285
912			\|a GBV_ILN_293
912			\|a GBV_ILN_370
912			\|a GBV_ILN_602
912			\|a GBV_ILN_636
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
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912			\|a GBV_ILN_2031
912			\|a GBV_ILN_2034
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912			\|a GBV_ILN_2050
912			\|a GBV_ILN_2055
912			\|a GBV_ILN_2057
912			\|a GBV_ILN_2059
912			\|a GBV_ILN_2061
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912			\|a GBV_ILN_2065
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912			\|a GBV_ILN_2070
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912			\|a GBV_ILN_2118
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matchkey_str	article:14321823:2016----::uosurseghf
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allfields	10.1007/s00209-016-1773-z doi (DE-627)SPR001930575 (SPR)s00209-016-1773-z-e DE-627 ger DE-627 rakwb eng Scheiderer, Claus verfasserin aut Sum of squares length of real forms 2016 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © Springer-Verlag Berlin Heidelberg 2016 Abstract For %$n,\,d\ge 1%$ let p(n, 2d) denote the smallest number p such that every sum of squares of degree d forms in %${\mathbb {R}}[x_1,\ldots ,x_n]%$ is a sum of p squares. We establish lower bounds for p(n, 2d) that are considerably stronger than the bounds known so far. Combined with known upper bounds they give %$p(3,2d)\in \{d+1,\,d+2\}%$ in the ternary case. Assuming a conjecture of Iarrobino–Kanev on dimensions of tangent spaces to catalecticant varieties, we show that %$p(n,2d)\sim const\cdot d^{(n-1)/2}%$ for %$d\rightarrow \infty %$ and all %$n\ge 3%$. For ternary sextics and quaternary quartics we determine the exact value of the invariant, showing %$p(3,6)=4%$ and %$p(4,4)=5%$. Lower Bound (dpeaa)DE-He213 Homogeneous Polynomial (dpeaa)DE-He213 Hilbert Series (dpeaa)DE-He213 Hilbert Function (dpeaa)DE-He213 Artinian Ring (dpeaa)DE-He213 Enthalten in Mathematische Zeitschrift Berlin : Springer, 1918 286(2016), 1-2 vom: 27. Okt., Seite 559-570 (DE-627)254630812 (DE-600)1462134-4 1432-1823 nnns volume:286 year:2016 number:1-2 day:27 month:10 pages:559-570 https://dx.doi.org/10.1007/s00209-016-1773-z lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER 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_105 GBV_ILN_110 GBV_ILN_120 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_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_267 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 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_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2070 GBV_ILN_2086 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_2188 GBV_ILN_2190 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_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_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_4393 GBV_ILN_4700 AR 286 2016 1-2 27 10 559-570
spelling	10.1007/s00209-016-1773-z doi (DE-627)SPR001930575 (SPR)s00209-016-1773-z-e DE-627 ger DE-627 rakwb eng Scheiderer, Claus verfasserin aut Sum of squares length of real forms 2016 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © Springer-Verlag Berlin Heidelberg 2016 Abstract For %$n,\,d\ge 1%$ let p(n, 2d) denote the smallest number p such that every sum of squares of degree d forms in %${\mathbb {R}}[x_1,\ldots ,x_n]%$ is a sum of p squares. We establish lower bounds for p(n, 2d) that are considerably stronger than the bounds known so far. Combined with known upper bounds they give %$p(3,2d)\in \{d+1,\,d+2\}%$ in the ternary case. Assuming a conjecture of Iarrobino–Kanev on dimensions of tangent spaces to catalecticant varieties, we show that %$p(n,2d)\sim const\cdot d^{(n-1)/2}%$ for %$d\rightarrow \infty %$ and all %$n\ge 3%$. For ternary sextics and quaternary quartics we determine the exact value of the invariant, showing %$p(3,6)=4%$ and %$p(4,4)=5%$. Lower Bound (dpeaa)DE-He213 Homogeneous Polynomial (dpeaa)DE-He213 Hilbert Series (dpeaa)DE-He213 Hilbert Function (dpeaa)DE-He213 Artinian Ring (dpeaa)DE-He213 Enthalten in Mathematische Zeitschrift Berlin : Springer, 1918 286(2016), 1-2 vom: 27. Okt., Seite 559-570 (DE-627)254630812 (DE-600)1462134-4 1432-1823 nnns volume:286 year:2016 number:1-2 day:27 month:10 pages:559-570 https://dx.doi.org/10.1007/s00209-016-1773-z lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER 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_105 GBV_ILN_110 GBV_ILN_120 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_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_267 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 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_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2070 GBV_ILN_2086 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_2188 GBV_ILN_2190 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_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_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_4393 GBV_ILN_4700 AR 286 2016 1-2 27 10 559-570
allfields_unstemmed	10.1007/s00209-016-1773-z doi (DE-627)SPR001930575 (SPR)s00209-016-1773-z-e DE-627 ger DE-627 rakwb eng Scheiderer, Claus verfasserin aut Sum of squares length of real forms 2016 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © Springer-Verlag Berlin Heidelberg 2016 Abstract For %$n,\,d\ge 1%$ let p(n, 2d) denote the smallest number p such that every sum of squares of degree d forms in %${\mathbb {R}}[x_1,\ldots ,x_n]%$ is a sum of p squares. We establish lower bounds for p(n, 2d) that are considerably stronger than the bounds known so far. Combined with known upper bounds they give %$p(3,2d)\in \{d+1,\,d+2\}%$ in the ternary case. Assuming a conjecture of Iarrobino–Kanev on dimensions of tangent spaces to catalecticant varieties, we show that %$p(n,2d)\sim const\cdot d^{(n-1)/2}%$ for %$d\rightarrow \infty %$ and all %$n\ge 3%$. For ternary sextics and quaternary quartics we determine the exact value of the invariant, showing %$p(3,6)=4%$ and %$p(4,4)=5%$. Lower Bound (dpeaa)DE-He213 Homogeneous Polynomial (dpeaa)DE-He213 Hilbert Series (dpeaa)DE-He213 Hilbert Function (dpeaa)DE-He213 Artinian Ring (dpeaa)DE-He213 Enthalten in Mathematische Zeitschrift Berlin : Springer, 1918 286(2016), 1-2 vom: 27. Okt., Seite 559-570 (DE-627)254630812 (DE-600)1462134-4 1432-1823 nnns volume:286 year:2016 number:1-2 day:27 month:10 pages:559-570 https://dx.doi.org/10.1007/s00209-016-1773-z lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER 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_105 GBV_ILN_110 GBV_ILN_120 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_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_267 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 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_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2070 GBV_ILN_2086 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_2188 GBV_ILN_2190 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_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_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_4393 GBV_ILN_4700 AR 286 2016 1-2 27 10 559-570
allfieldsGer	10.1007/s00209-016-1773-z doi (DE-627)SPR001930575 (SPR)s00209-016-1773-z-e DE-627 ger DE-627 rakwb eng Scheiderer, Claus verfasserin aut Sum of squares length of real forms 2016 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © Springer-Verlag Berlin Heidelberg 2016 Abstract For %$n,\,d\ge 1%$ let p(n, 2d) denote the smallest number p such that every sum of squares of degree d forms in %${\mathbb {R}}[x_1,\ldots ,x_n]%$ is a sum of p squares. We establish lower bounds for p(n, 2d) that are considerably stronger than the bounds known so far. Combined with known upper bounds they give %$p(3,2d)\in \{d+1,\,d+2\}%$ in the ternary case. Assuming a conjecture of Iarrobino–Kanev on dimensions of tangent spaces to catalecticant varieties, we show that %$p(n,2d)\sim const\cdot d^{(n-1)/2}%$ for %$d\rightarrow \infty %$ and all %$n\ge 3%$. For ternary sextics and quaternary quartics we determine the exact value of the invariant, showing %$p(3,6)=4%$ and %$p(4,4)=5%$. Lower Bound (dpeaa)DE-He213 Homogeneous Polynomial (dpeaa)DE-He213 Hilbert Series (dpeaa)DE-He213 Hilbert Function (dpeaa)DE-He213 Artinian Ring (dpeaa)DE-He213 Enthalten in Mathematische Zeitschrift Berlin : Springer, 1918 286(2016), 1-2 vom: 27. Okt., Seite 559-570 (DE-627)254630812 (DE-600)1462134-4 1432-1823 nnns volume:286 year:2016 number:1-2 day:27 month:10 pages:559-570 https://dx.doi.org/10.1007/s00209-016-1773-z lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER 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_105 GBV_ILN_110 GBV_ILN_120 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_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_267 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 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_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2070 GBV_ILN_2086 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_2188 GBV_ILN_2190 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_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_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_4393 GBV_ILN_4700 AR 286 2016 1-2 27 10 559-570
allfieldsSound	10.1007/s00209-016-1773-z doi (DE-627)SPR001930575 (SPR)s00209-016-1773-z-e DE-627 ger DE-627 rakwb eng Scheiderer, Claus verfasserin aut Sum of squares length of real forms 2016 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © Springer-Verlag Berlin Heidelberg 2016 Abstract For %$n,\,d\ge 1%$ let p(n, 2d) denote the smallest number p such that every sum of squares of degree d forms in %${\mathbb {R}}[x_1,\ldots ,x_n]%$ is a sum of p squares. We establish lower bounds for p(n, 2d) that are considerably stronger than the bounds known so far. Combined with known upper bounds they give %$p(3,2d)\in \{d+1,\,d+2\}%$ in the ternary case. Assuming a conjecture of Iarrobino–Kanev on dimensions of tangent spaces to catalecticant varieties, we show that %$p(n,2d)\sim const\cdot d^{(n-1)/2}%$ for %$d\rightarrow \infty %$ and all %$n\ge 3%$. For ternary sextics and quaternary quartics we determine the exact value of the invariant, showing %$p(3,6)=4%$ and %$p(4,4)=5%$. Lower Bound (dpeaa)DE-He213 Homogeneous Polynomial (dpeaa)DE-He213 Hilbert Series (dpeaa)DE-He213 Hilbert Function (dpeaa)DE-He213 Artinian Ring (dpeaa)DE-He213 Enthalten in Mathematische Zeitschrift Berlin : Springer, 1918 286(2016), 1-2 vom: 27. Okt., Seite 559-570 (DE-627)254630812 (DE-600)1462134-4 1432-1823 nnns volume:286 year:2016 number:1-2 day:27 month:10 pages:559-570 https://dx.doi.org/10.1007/s00209-016-1773-z lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER 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_105 GBV_ILN_110 GBV_ILN_120 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_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_267 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 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_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2070 GBV_ILN_2086 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_2188 GBV_ILN_2190 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_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_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_4393 GBV_ILN_4700 AR 286 2016 1-2 27 10 559-570
language	English
source	Enthalten in Mathematische Zeitschrift 286(2016), 1-2 vom: 27. Okt., Seite 559-570 volume:286 year:2016 number:1-2 day:27 month:10 pages:559-570
sourceStr	Enthalten in Mathematische Zeitschrift 286(2016), 1-2 vom: 27. Okt., Seite 559-570 volume:286 year:2016 number:1-2 day:27 month:10 pages:559-570
format_phy_str_mv	Article
institution	findex.gbv.de
topic_facet	Lower Bound Homogeneous Polynomial Hilbert Series Hilbert Function Artinian Ring
isfreeaccess_bool	false
container_title	Mathematische Zeitschrift
authorswithroles_txt_mv	Scheiderer, Claus @@aut@@
publishDateDaySort_date	2016-10-27T00:00:00Z
hierarchy_top_id	254630812
id	SPR001930575
language_de	englisch
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We establish lower bounds for p(n, 2d) that are considerably stronger than the bounds known so far. Combined with known upper bounds they give %$p(3,2d)\in \{d+1,\,d+2\}%$ in the ternary case. Assuming a conjecture of Iarrobino–Kanev on dimensions of tangent spaces to catalecticant varieties, we show that %$p(n,2d)\sim const\cdot d^{(n-1)/2}%$ for %$d\rightarrow \infty %$ and all %$n\ge 3%$. 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author	Scheiderer, Claus
spellingShingle	Scheiderer, Claus misc Lower Bound misc Homogeneous Polynomial misc Hilbert Series misc Hilbert Function misc Artinian Ring Sum of squares length of real forms
authorStr	Scheiderer, Claus
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topic_title	Sum of squares length of real forms Lower Bound (dpeaa)DE-He213 Homogeneous Polynomial (dpeaa)DE-He213 Hilbert Series (dpeaa)DE-He213 Hilbert Function (dpeaa)DE-He213 Artinian Ring (dpeaa)DE-He213
topic	misc Lower Bound misc Homogeneous Polynomial misc Hilbert Series misc Hilbert Function misc Artinian Ring
topic_unstemmed	misc Lower Bound misc Homogeneous Polynomial misc Hilbert Series misc Hilbert Function misc Artinian Ring
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title	Sum of squares length of real forms
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title_full	Sum of squares length of real forms
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title_sort	sum of squares length of real forms
title_auth	Sum of squares length of real forms
abstract	Abstract For %$n,\,d\ge 1%$ let p(n, 2d) denote the smallest number p such that every sum of squares of degree d forms in %${\mathbb {R}}[x_1,\ldots ,x_n]%$ is a sum of p squares. We establish lower bounds for p(n, 2d) that are considerably stronger than the bounds known so far. Combined with known upper bounds they give %$p(3,2d)\in \{d+1,\,d+2\}%$ in the ternary case. Assuming a conjecture of Iarrobino–Kanev on dimensions of tangent spaces to catalecticant varieties, we show that %$p(n,2d)\sim const\cdot d^{(n-1)/2}%$ for %$d\rightarrow \infty %$ and all %$n\ge 3%$. For ternary sextics and quaternary quartics we determine the exact value of the invariant, showing %$p(3,6)=4%$ and %$p(4,4)=5%$. © Springer-Verlag Berlin Heidelberg 2016
abstractGer	Abstract For %$n,\,d\ge 1%$ let p(n, 2d) denote the smallest number p such that every sum of squares of degree d forms in %${\mathbb {R}}[x_1,\ldots ,x_n]%$ is a sum of p squares. We establish lower bounds for p(n, 2d) that are considerably stronger than the bounds known so far. Combined with known upper bounds they give %$p(3,2d)\in \{d+1,\,d+2\}%$ in the ternary case. Assuming a conjecture of Iarrobino–Kanev on dimensions of tangent spaces to catalecticant varieties, we show that %$p(n,2d)\sim const\cdot d^{(n-1)/2}%$ for %$d\rightarrow \infty %$ and all %$n\ge 3%$. For ternary sextics and quaternary quartics we determine the exact value of the invariant, showing %$p(3,6)=4%$ and %$p(4,4)=5%$. © Springer-Verlag Berlin Heidelberg 2016
abstract_unstemmed	Abstract For %$n,\,d\ge 1%$ let p(n, 2d) denote the smallest number p such that every sum of squares of degree d forms in %${\mathbb {R}}[x_1,\ldots ,x_n]%$ is a sum of p squares. We establish lower bounds for p(n, 2d) that are considerably stronger than the bounds known so far. Combined with known upper bounds they give %$p(3,2d)\in \{d+1,\,d+2\}%$ in the ternary case. Assuming a conjecture of Iarrobino–Kanev on dimensions of tangent spaces to catalecticant varieties, we show that %$p(n,2d)\sim const\cdot d^{(n-1)/2}%$ for %$d\rightarrow \infty %$ and all %$n\ge 3%$. For ternary sextics and quaternary quartics we determine the exact value of the invariant, showing %$p(3,6)=4%$ and %$p(4,4)=5%$. © Springer-Verlag Berlin Heidelberg 2016
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container_issue	1-2
title_short	Sum of squares length of real forms
url	https://dx.doi.org/10.1007/s00209-016-1773-z
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doi_str	10.1007/s00209-016-1773-z
up_date	2024-07-04T01:02:05.880Z
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