Łojasiewicz exponents and Farey sequences
Let I be an ideal of the ring of formal power series K [ [ x , y ] ] with coefficients in an algebraically closed field K of arbitrary characteristic. Let Φ denote the set of all parametrizations φ = ( φ 1 , φ 2 ) ∈ K [ [ t ] ] 2 , where φ ≠ ( 0 , 0 ) and φ ( 0 , 0 ) = ( 0 , 0 ) . The purpose of thi...
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| Veröffentlicht in: | Revista matemática complutense 2016-09, Vol.29 (3), p.719-724 |
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| creator | de Felipe, A. B. García Barroso, E. R. Gwoździewicz, J. Płoski, A. |
| description | Let
I
be an ideal of the ring of formal power series
K
[
[
x
,
y
]
]
with coefficients in an algebraically closed field
K
of arbitrary characteristic. Let
Φ
denote the set of all parametrizations
φ
=
(
φ
1
,
φ
2
)
∈
K
[
[
t
]
]
2
, where
φ
≠
(
0
,
0
)
and
φ
(
0
,
0
)
=
(
0
,
0
)
. The purpose of this paper is to investigate the invariant
L
0
(
I
)
=
sup
φ
∈
Φ
inf
f
∈
I
ord
f
∘
φ
ord
φ
called the
Łojasiewicz exponent
of
I
. Our main result states that for the ideals
I
of finite codimension the Łojasiewicz exponent
L
0
(
I
)
is a Farey number i.e. an integer or a rational number of the form
N
+
b
a
, where
a
,
b
,
N
are integers such that
0
<
b
<
a
<
N
. |
| doi_str_mv | 10.1007/s13163-016-0194-1 |
| format | Article |
| fullrecord | <record><control><sourceid>proquest_cross</sourceid><recordid>TN_cdi_proquest_journals_1880882906</recordid><sourceformat>XML</sourceformat><sourcesystem>PC</sourcesystem><sourcerecordid>1880882906</sourcerecordid><originalsourceid>FETCH-LOGICAL-c268t-aeece1318657124f8cc9ff69bfa7055997738ea5f3da1e328a96e85d714344793</originalsourceid><addsrcrecordid>eNp1kM9KAzEQxoMoWKsP4G3BmxDNbLLJ5CjFqlDwoucQ04nsors1adF68-F8L1PWgxcP8-fwffMNP8ZOQVyAEOYygwQtuQBdyioOe2wCFpHXKMx-2UFaXhoesqOcOyEaq1BN2Pn319D53NJ7Gz4r-lgNPfXrXPl-Wc19om2V6W1DfaB8zA6if8l08jun7HF-_TC75Yv7m7vZ1YKHWuOae6JA5RvUjYFaRQzBxqjtU_RGNI21xkgk30S59ECyRm81YbM0oKRSxsopOxvvrtJQovPadcMm9SXSAaJArK3QRQWjKqQh50TRrVL76tPWgXA7JG5E4goSt0PioHjq0ZOLtn-m9Ofyv6Yf-_Zi2g</addsrcrecordid><sourcetype>Aggregation Database</sourcetype><iscdi>true</iscdi><recordtype>article</recordtype><pqid>1880882906</pqid></control><display><type>article</type><title>Łojasiewicz exponents and Farey sequences</title><source>Universidad Complutense de Madrid Free Journals</source><source>SpringerLink Journals - AutoHoldings</source><creator>de Felipe, A. B. ; García Barroso, E. R. ; Gwoździewicz, J. ; Płoski, A.</creator><creatorcontrib>de Felipe, A. B. ; García Barroso, E. R. ; Gwoździewicz, J. ; Płoski, A.</creatorcontrib><description>Let
I
be an ideal of the ring of formal power series
K
[
[
x
,
y
]
]
with coefficients in an algebraically closed field
K
of arbitrary characteristic. Let
Φ
denote the set of all parametrizations
φ
=
(
φ
1
,
φ
2
)
∈
K
[
[
t
]
]
2
, where
φ
≠
(
0
,
0
)
and
φ
(
0
,
0
)
=
(
0
,
0
)
. The purpose of this paper is to investigate the invariant
L
0
(
I
)
=
sup
φ
∈
Φ
inf
f
∈
I
ord
f
∘
φ
ord
φ
called the
Łojasiewicz exponent
of
I
. Our main result states that for the ideals
I
of finite codimension the Łojasiewicz exponent
L
0
(
I
)
is a Farey number i.e. an integer or a rational number of the form
N
+
b
a
, where
a
,
b
,
N
are integers such that
0
<
b
<
a
<
N
.</description><identifier>ISSN: 1139-1138</identifier><identifier>EISSN: 1988-2807</identifier><identifier>DOI: 10.1007/s13163-016-0194-1</identifier><language>eng</language><publisher>Milan: Springer Milan</publisher><subject>Algebra ; Analysis ; Applications of Mathematics ; Geometry ; Integers ; Mathematics ; Mathematics and Statistics ; Power series ; Sequences ; Topology</subject><ispartof>Revista matemática complutense, 2016-09, Vol.29 (3), p.719-724</ispartof><rights>European union 2016</rights><rights>Copyright Springer Science & Business Media 2016</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><cites>FETCH-LOGICAL-c268t-aeece1318657124f8cc9ff69bfa7055997738ea5f3da1e328a96e85d714344793</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktopdf>$$Uhttps://link.springer.com/content/pdf/10.1007/s13163-016-0194-1$$EPDF$$P50$$Gspringer$$H</linktopdf><linktohtml>$$Uhttps://link.springer.com/10.1007/s13163-016-0194-1$$EHTML$$P50$$Gspringer$$H</linktohtml><link.rule.ids>314,780,784,27924,27925,41488,42557,51319</link.rule.ids></links><search><creatorcontrib>de Felipe, A. B.</creatorcontrib><creatorcontrib>García Barroso, E. R.</creatorcontrib><creatorcontrib>Gwoździewicz, J.</creatorcontrib><creatorcontrib>Płoski, A.</creatorcontrib><title>Łojasiewicz exponents and Farey sequences</title><title>Revista matemática complutense</title><addtitle>Rev Mat Complut</addtitle><description>Let
I
be an ideal of the ring of formal power series
K
[
[
x
,
y
]
]
with coefficients in an algebraically closed field
K
of arbitrary characteristic. Let
Φ
denote the set of all parametrizations
φ
=
(
φ
1
,
φ
2
)
∈
K
[
[
t
]
]
2
, where
φ
≠
(
0
,
0
)
and
φ
(
0
,
0
)
=
(
0
,
0
)
. The purpose of this paper is to investigate the invariant
L
0
(
I
)
=
sup
φ
∈
Φ
inf
f
∈
I
ord
f
∘
φ
ord
φ
called the
Łojasiewicz exponent
of
I
. Our main result states that for the ideals
I
of finite codimension the Łojasiewicz exponent
L
0
(
I
)
is a Farey number i.e. an integer or a rational number of the form
N
+
b
a
, where
a
,
b
,
N
are integers such that
0
<
b
<
a
<
N
.</description><subject>Algebra</subject><subject>Analysis</subject><subject>Applications of Mathematics</subject><subject>Geometry</subject><subject>Integers</subject><subject>Mathematics</subject><subject>Mathematics and Statistics</subject><subject>Power series</subject><subject>Sequences</subject><subject>Topology</subject><issn>1139-1138</issn><issn>1988-2807</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2016</creationdate><recordtype>article</recordtype><recordid>eNp1kM9KAzEQxoMoWKsP4G3BmxDNbLLJ5CjFqlDwoucQ04nsors1adF68-F8L1PWgxcP8-fwffMNP8ZOQVyAEOYygwQtuQBdyioOe2wCFpHXKMx-2UFaXhoesqOcOyEaq1BN2Pn319D53NJ7Gz4r-lgNPfXrXPl-Wc19om2V6W1DfaB8zA6if8l08jun7HF-_TC75Yv7m7vZ1YKHWuOae6JA5RvUjYFaRQzBxqjtU_RGNI21xkgk30S59ECyRm81YbM0oKRSxsopOxvvrtJQovPadcMm9SXSAaJArK3QRQWjKqQh50TRrVL76tPWgXA7JG5E4goSt0PioHjq0ZOLtn-m9Ofyv6Yf-_Zi2g</recordid><startdate>20160901</startdate><enddate>20160901</enddate><creator>de Felipe, A. B.</creator><creator>García Barroso, E. R.</creator><creator>Gwoździewicz, J.</creator><creator>Płoski, A.</creator><general>Springer Milan</general><general>Springer Nature B.V</general><scope>AAYXX</scope><scope>CITATION</scope></search><sort><creationdate>20160901</creationdate><title>Łojasiewicz exponents and Farey sequences</title><author>de Felipe, A. B. ; García Barroso, E. R. ; Gwoździewicz, J. ; Płoski, A.</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c268t-aeece1318657124f8cc9ff69bfa7055997738ea5f3da1e328a96e85d714344793</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2016</creationdate><topic>Algebra</topic><topic>Analysis</topic><topic>Applications of Mathematics</topic><topic>Geometry</topic><topic>Integers</topic><topic>Mathematics</topic><topic>Mathematics and Statistics</topic><topic>Power series</topic><topic>Sequences</topic><topic>Topology</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>de Felipe, A. B.</creatorcontrib><creatorcontrib>García Barroso, E. R.</creatorcontrib><creatorcontrib>Gwoździewicz, J.</creatorcontrib><creatorcontrib>Płoski, A.</creatorcontrib><collection>CrossRef</collection><jtitle>Revista matemática complutense</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>de Felipe, A. B.</au><au>García Barroso, E. R.</au><au>Gwoździewicz, J.</au><au>Płoski, A.</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Łojasiewicz exponents and Farey sequences</atitle><jtitle>Revista matemática complutense</jtitle><stitle>Rev Mat Complut</stitle><date>2016-09-01</date><risdate>2016</risdate><volume>29</volume><issue>3</issue><spage>719</spage><epage>724</epage><pages>719-724</pages><issn>1139-1138</issn><eissn>1988-2807</eissn><abstract>Let
I
be an ideal of the ring of formal power series
K
[
[
x
,
y
]
]
with coefficients in an algebraically closed field
K
of arbitrary characteristic. Let
Φ
denote the set of all parametrizations
φ
=
(
φ
1
,
φ
2
)
∈
K
[
[
t
]
]
2
, where
φ
≠
(
0
,
0
)
and
φ
(
0
,
0
)
=
(
0
,
0
)
. The purpose of this paper is to investigate the invariant
L
0
(
I
)
=
sup
φ
∈
Φ
inf
f
∈
I
ord
f
∘
φ
ord
φ
called the
Łojasiewicz exponent
of
I
. Our main result states that for the ideals
I
of finite codimension the Łojasiewicz exponent
L
0
(
I
)
is a Farey number i.e. an integer or a rational number of the form
N
+
b
a
, where
a
,
b
,
N
are integers such that
0
<
b
<
a
<
N
.</abstract><cop>Milan</cop><pub>Springer Milan</pub><doi>10.1007/s13163-016-0194-1</doi><tpages>6</tpages></addata></record> |
| fulltext | fulltext |
| identifier | ISSN: 1139-1138 |
| ispartof | Revista matemática complutense, 2016-09, Vol.29 (3), p.719-724 |
| issn | 1139-1138 1988-2807 |
| language | eng |
| recordid | cdi_proquest_journals_1880882906 |
| source | Universidad Complutense de Madrid Free Journals; SpringerLink Journals - AutoHoldings |
| subjects | Algebra Analysis Applications of Mathematics Geometry Integers Mathematics Mathematics and Statistics Power series Sequences Topology |
| title | Łojasiewicz exponents and Farey sequences |
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