Polynomial matrices, splitting subspaces and Krylov subspaces over finite fields

Let T be a linear operator on a vector space V of dimension n over F q . For any divisor m of n, an m-dimensional subsp...
Ausführliche Beschreibung

Let T be a linear operator on a vector space V of dimension n over F q . For any divisor m of n, an m-dimensional subspace W of V is T-splitting if V = W ⊕ T W ⊕ ⋯ ⊕ T d − 1 W , where d = n / m . Let σ ( m , d ; T ) denote the number of m-dimensional T-splitting subspaces. Determining σ ( m , d ; T ) for an arbitrary operator T is an open problem that is closely related to another important problem on Krylov spaces. We discuss this connection and give explicit formulae for σ ( m , d ; T ) in the case where the invariant factors of T satisfy certain degree conditions. A connection with an enumeration problem on polynomial matrices is also discussed. Ausführliche Beschreibung

Gespeichert in:

Autor*in:	Aggarwal, Divya [verfasserIn] Ram, Samrith [verfasserIn]

Format:	E-Artikel
Sprache:	Englisch

Erschienen:	2022

Schlagwörter:	Splitting subspace Krylov space Antiinvariant subspace Polynomial matrix Invariant factor Finite field

Übergeordnetes Werk:	Enthalten in: No title available - 83
Übergeordnetes Werk:	volume:83

DOI / URN:	10.1016/j.ffa.2022.102081

Katalog-ID:	ELV008338868

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520			\|a Let T be a linear operator on a vector space V of dimension n over F q . For any divisor m of n, an m-dimensional subspace W of V is T-splitting if V = W ⊕ T W ⊕ ⋯ ⊕ T d − 1 W , where d = n / m . Let σ ( m , d ; T ) denote the number of m-dimensional T-splitting subspaces. Determining σ ( m , d ; T ) for an arbitrary operator T is an open problem that is closely related to another important problem on Krylov spaces. We discuss this connection and give explicit formulae for σ ( m , d ; T ) in the case where the invariant factors of T satisfy certain degree conditions. A connection with an enumeration problem on polynomial matrices is also discussed.
650		4	\|a Splitting subspace
650		4	\|a Krylov space
650		4	\|a Antiinvariant subspace
650		4	\|a Polynomial matrix
650		4	\|a Invariant factor
650		4	\|a Finite field
700	1		\|a Ram, Samrith \|e verfasserin \|0 (orcid)0000-0001-5870-3452 \|4 aut
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allfields	10.1016/j.ffa.2022.102081 doi (DE-627)ELV008338868 (ELSEVIER)S1071-5797(22)00090-9 DE-627 ger DE-627 rda eng Aggarwal, Divya verfasserin aut Polynomial matrices, splitting subspaces and Krylov subspaces over finite fields 2022 nicht spezifiziert zzz rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier Let T be a linear operator on a vector space V of dimension n over F q . For any divisor m of n, an m-dimensional subspace W of V is T-splitting if V = W ⊕ T W ⊕ ⋯ ⊕ T d − 1 W , where d = n / m . Let σ ( m , d ; T ) denote the number of m-dimensional T-splitting subspaces. Determining σ ( m , d ; T ) for an arbitrary operator T is an open problem that is closely related to another important problem on Krylov spaces. We discuss this connection and give explicit formulae for σ ( m , d ; T ) in the case where the invariant factors of T satisfy certain degree conditions. A connection with an enumeration problem on polynomial matrices is also discussed. Splitting subspace Krylov space Antiinvariant subspace Polynomial matrix Invariant factor Finite field Ram, Samrith verfasserin (orcid)0000-0001-5870-3452 aut Enthalten in No title available 83 (DE-627)26687701X 1071-5797 nnns volume:83 GBV_USEFLAG_U SYSFLAG_U GBV_ELV GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 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_150 GBV_ILN_151 GBV_ILN_224 GBV_ILN_370 GBV_ILN_602 GBV_ILN_702 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2027 GBV_ILN_2034 GBV_ILN_2038 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2056 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2118 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2190 GBV_ILN_2336 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_4012 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 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_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4338 GBV_ILN_4367 GBV_ILN_4393 GBV_ILN_4700 AR 83
spelling	10.1016/j.ffa.2022.102081 doi (DE-627)ELV008338868 (ELSEVIER)S1071-5797(22)00090-9 DE-627 ger DE-627 rda eng Aggarwal, Divya verfasserin aut Polynomial matrices, splitting subspaces and Krylov subspaces over finite fields 2022 nicht spezifiziert zzz rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier Let T be a linear operator on a vector space V of dimension n over F q . For any divisor m of n, an m-dimensional subspace W of V is T-splitting if V = W ⊕ T W ⊕ ⋯ ⊕ T d − 1 W , where d = n / m . Let σ ( m , d ; T ) denote the number of m-dimensional T-splitting subspaces. Determining σ ( m , d ; T ) for an arbitrary operator T is an open problem that is closely related to another important problem on Krylov spaces. We discuss this connection and give explicit formulae for σ ( m , d ; T ) in the case where the invariant factors of T satisfy certain degree conditions. A connection with an enumeration problem on polynomial matrices is also discussed. Splitting subspace Krylov space Antiinvariant subspace Polynomial matrix Invariant factor Finite field Ram, Samrith verfasserin (orcid)0000-0001-5870-3452 aut Enthalten in No title available 83 (DE-627)26687701X 1071-5797 nnns volume:83 GBV_USEFLAG_U SYSFLAG_U GBV_ELV GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 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_150 GBV_ILN_151 GBV_ILN_224 GBV_ILN_370 GBV_ILN_602 GBV_ILN_702 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2027 GBV_ILN_2034 GBV_ILN_2038 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2056 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2118 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2190 GBV_ILN_2336 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_4012 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 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_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4338 GBV_ILN_4367 GBV_ILN_4393 GBV_ILN_4700 AR 83
allfields_unstemmed	10.1016/j.ffa.2022.102081 doi (DE-627)ELV008338868 (ELSEVIER)S1071-5797(22)00090-9 DE-627 ger DE-627 rda eng Aggarwal, Divya verfasserin aut Polynomial matrices, splitting subspaces and Krylov subspaces over finite fields 2022 nicht spezifiziert zzz rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier Let T be a linear operator on a vector space V of dimension n over F q . For any divisor m of n, an m-dimensional subspace W of V is T-splitting if V = W ⊕ T W ⊕ ⋯ ⊕ T d − 1 W , where d = n / m . Let σ ( m , d ; T ) denote the number of m-dimensional T-splitting subspaces. Determining σ ( m , d ; T ) for an arbitrary operator T is an open problem that is closely related to another important problem on Krylov spaces. We discuss this connection and give explicit formulae for σ ( m , d ; T ) in the case where the invariant factors of T satisfy certain degree conditions. A connection with an enumeration problem on polynomial matrices is also discussed. Splitting subspace Krylov space Antiinvariant subspace Polynomial matrix Invariant factor Finite field Ram, Samrith verfasserin (orcid)0000-0001-5870-3452 aut Enthalten in No title available 83 (DE-627)26687701X 1071-5797 nnns volume:83 GBV_USEFLAG_U SYSFLAG_U GBV_ELV GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 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_150 GBV_ILN_151 GBV_ILN_224 GBV_ILN_370 GBV_ILN_602 GBV_ILN_702 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2027 GBV_ILN_2034 GBV_ILN_2038 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2056 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2118 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2190 GBV_ILN_2336 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_4012 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 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_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4338 GBV_ILN_4367 GBV_ILN_4393 GBV_ILN_4700 AR 83
allfieldsGer	10.1016/j.ffa.2022.102081 doi (DE-627)ELV008338868 (ELSEVIER)S1071-5797(22)00090-9 DE-627 ger DE-627 rda eng Aggarwal, Divya verfasserin aut Polynomial matrices, splitting subspaces and Krylov subspaces over finite fields 2022 nicht spezifiziert zzz rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier Let T be a linear operator on a vector space V of dimension n over F q . For any divisor m of n, an m-dimensional subspace W of V is T-splitting if V = W ⊕ T W ⊕ ⋯ ⊕ T d − 1 W , where d = n / m . Let σ ( m , d ; T ) denote the number of m-dimensional T-splitting subspaces. Determining σ ( m , d ; T ) for an arbitrary operator T is an open problem that is closely related to another important problem on Krylov spaces. We discuss this connection and give explicit formulae for σ ( m , d ; T ) in the case where the invariant factors of T satisfy certain degree conditions. A connection with an enumeration problem on polynomial matrices is also discussed. Splitting subspace Krylov space Antiinvariant subspace Polynomial matrix Invariant factor Finite field Ram, Samrith verfasserin (orcid)0000-0001-5870-3452 aut Enthalten in No title available 83 (DE-627)26687701X 1071-5797 nnns volume:83 GBV_USEFLAG_U SYSFLAG_U GBV_ELV GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 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_150 GBV_ILN_151 GBV_ILN_224 GBV_ILN_370 GBV_ILN_602 GBV_ILN_702 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2027 GBV_ILN_2034 GBV_ILN_2038 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2056 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2118 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2190 GBV_ILN_2336 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_4012 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 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_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4338 GBV_ILN_4367 GBV_ILN_4393 GBV_ILN_4700 AR 83
allfieldsSound	10.1016/j.ffa.2022.102081 doi (DE-627)ELV008338868 (ELSEVIER)S1071-5797(22)00090-9 DE-627 ger DE-627 rda eng Aggarwal, Divya verfasserin aut Polynomial matrices, splitting subspaces and Krylov subspaces over finite fields 2022 nicht spezifiziert zzz rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier Let T be a linear operator on a vector space V of dimension n over F q . For any divisor m of n, an m-dimensional subspace W of V is T-splitting if V = W ⊕ T W ⊕ ⋯ ⊕ T d − 1 W , where d = n / m . Let σ ( m , d ; T ) denote the number of m-dimensional T-splitting subspaces. Determining σ ( m , d ; T ) for an arbitrary operator T is an open problem that is closely related to another important problem on Krylov spaces. We discuss this connection and give explicit formulae for σ ( m , d ; T ) in the case where the invariant factors of T satisfy certain degree conditions. A connection with an enumeration problem on polynomial matrices is also discussed. Splitting subspace Krylov space Antiinvariant subspace Polynomial matrix Invariant factor Finite field Ram, Samrith verfasserin (orcid)0000-0001-5870-3452 aut Enthalten in No title available 83 (DE-627)26687701X 1071-5797 nnns volume:83 GBV_USEFLAG_U SYSFLAG_U GBV_ELV GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 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_150 GBV_ILN_151 GBV_ILN_224 GBV_ILN_370 GBV_ILN_602 GBV_ILN_702 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2027 GBV_ILN_2034 GBV_ILN_2038 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2056 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2118 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2190 GBV_ILN_2336 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_4012 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 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_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4338 GBV_ILN_4367 GBV_ILN_4393 GBV_ILN_4700 AR 83
language	English
source	Enthalten in No title available 83 volume:83
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For any divisor m of n, an m-dimensional subspace W of V is T-splitting if V = W ⊕ T W ⊕ ⋯ ⊕ T d − 1 W , where d = n / m . Let σ ( m , d ; T ) denote the number of m-dimensional T-splitting subspaces. Determining σ ( m , d ; T ) for an arbitrary operator T is an open problem that is closely related to another important problem on Krylov spaces. We discuss this connection and give explicit formulae for σ ( m , d ; T ) in the case where the invariant factors of T satisfy certain degree conditions. 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author	Aggarwal, Divya
spellingShingle	Aggarwal, Divya misc Splitting subspace misc Krylov space misc Antiinvariant subspace misc Polynomial matrix misc Invariant factor misc Finite field Polynomial matrices, splitting subspaces and Krylov subspaces over finite fields
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topic_title	Polynomial matrices, splitting subspaces and Krylov subspaces over finite fields Splitting subspace Krylov space Antiinvariant subspace Polynomial matrix Invariant factor Finite field
topic	misc Splitting subspace misc Krylov space misc Antiinvariant subspace misc Polynomial matrix misc Invariant factor misc Finite field
topic_unstemmed	misc Splitting subspace misc Krylov space misc Antiinvariant subspace misc Polynomial matrix misc Invariant factor misc Finite field
topic_browse	misc Splitting subspace misc Krylov space misc Antiinvariant subspace misc Polynomial matrix misc Invariant factor misc Finite field
format_facet	Elektronische Aufsätze Aufsätze Elektronische Ressource
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title	Polynomial matrices, splitting subspaces and Krylov subspaces over finite fields
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title_full	Polynomial matrices, splitting subspaces and Krylov subspaces over finite fields
author_sort	Aggarwal, Divya
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author_browse	Aggarwal, Divya Ram, Samrith
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author-letter	Aggarwal, Divya
doi_str_mv	10.1016/j.ffa.2022.102081
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title_sort	polynomial matrices, splitting subspaces and krylov subspaces over finite fields
title_auth	Polynomial matrices, splitting subspaces and Krylov subspaces over finite fields
abstract	Let T be a linear operator on a vector space V of dimension n over F q . For any divisor m of n, an m-dimensional subspace W of V is T-splitting if V = W ⊕ T W ⊕ ⋯ ⊕ T d − 1 W , where d = n / m . Let σ ( m , d ; T ) denote the number of m-dimensional T-splitting subspaces. Determining σ ( m , d ; T ) for an arbitrary operator T is an open problem that is closely related to another important problem on Krylov spaces. We discuss this connection and give explicit formulae for σ ( m , d ; T ) in the case where the invariant factors of T satisfy certain degree conditions. A connection with an enumeration problem on polynomial matrices is also discussed.
abstractGer	Let T be a linear operator on a vector space V of dimension n over F q . For any divisor m of n, an m-dimensional subspace W of V is T-splitting if V = W ⊕ T W ⊕ ⋯ ⊕ T d − 1 W , where d = n / m . Let σ ( m , d ; T ) denote the number of m-dimensional T-splitting subspaces. Determining σ ( m , d ; T ) for an arbitrary operator T is an open problem that is closely related to another important problem on Krylov spaces. We discuss this connection and give explicit formulae for σ ( m , d ; T ) in the case where the invariant factors of T satisfy certain degree conditions. A connection with an enumeration problem on polynomial matrices is also discussed.
abstract_unstemmed	Let T be a linear operator on a vector space V of dimension n over F q . For any divisor m of n, an m-dimensional subspace W of V is T-splitting if V = W ⊕ T W ⊕ ⋯ ⊕ T d − 1 W , where d = n / m . Let σ ( m , d ; T ) denote the number of m-dimensional T-splitting subspaces. Determining σ ( m , d ; T ) for an arbitrary operator T is an open problem that is closely related to another important problem on Krylov spaces. We discuss this connection and give explicit formulae for σ ( m , d ; T ) in the case where the invariant factors of T satisfy certain degree conditions. A connection with an enumeration problem on polynomial matrices is also discussed.
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title_short	Polynomial matrices, splitting subspaces and Krylov subspaces over finite fields
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doi_str	10.1016/j.ffa.2022.102081
up_date	2024-07-06T19:21:46.114Z
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For any divisor m of n, an m-dimensional subspace W of V is T-splitting if V = W ⊕ T W ⊕ ⋯ ⊕ T d − 1 W , where d = n / m . Let σ ( m , d ; T ) denote the number of m-dimensional T-splitting subspaces. Determining σ ( m , d ; T ) for an arbitrary operator T is an open problem that is closely related to another important problem on Krylov spaces. We discuss this connection and give explicit formulae for σ ( m , d ; T ) in the case where the invariant factors of T satisfy certain degree conditions. 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Polynomial matrices, splitting subspaces and Krylov subspaces over finite fields

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