{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T18:50:50Z","timestamp":1776797450968,"version":"3.51.2"},"reference-count":15,"publisher":"American Mathematical Society (AMS)","issue":"294","license":[{"start":{"date-parts":[[2015,10,9]],"date-time":"2015-10-09T00:00:00Z","timestamp":1444348800000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    Given a rational projective parametrization\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"script upper P left-parenthesis German s comma German t comma German v right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">P<\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mi mathvariant=\"fraktur\">s<\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mi mathvariant=\"fraktur\">t<\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mi mathvariant=\"fraktur\">v<\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\mathcal {P}(\\mathfrak {s},\\mathfrak {t},\\mathfrak {v})<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    of a rational projective surface\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"script upper S\">\n                        <mml:semantics>\n                          <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                            <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">S<\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\mathcal {S}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    we present an algorithm such that, with the exception of a finite set (maybe empty)\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"German upper B\">\n                        <mml:semantics>\n                          <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                            <mml:mi mathvariant=\"fraktur\">B<\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\mathfrak {B}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    of projective base points of\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"script upper P\">\n                        <mml:semantics>\n                          <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                            <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">P<\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\mathcal {P}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , decomposes the projective parameter plane as\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"double-struck upper P squared left-parenthesis double-struck upper K right-parenthesis minus German upper B equals union Underscript k equals 1 Overscript script l Endscripts German upper S Subscript k\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:msup>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                  <mml:mi mathvariant=\"double-struck\">P<\/mml:mi>\n                                <\/mml:mrow>\n                              <\/mml:mrow>\n                              <mml:mn>2<\/mml:mn>\n                            <\/mml:msup>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mi mathvariant=\"double-struck\">K<\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <mml:mo class=\"MJX-variant\">\n                              \u2216\n                              \n                            <\/mml:mo>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mi mathvariant=\"fraktur\">B<\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:munderover>\n                              <mml:mo>\n                                \u22c3\n                                \n                              <\/mml:mo>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi>k<\/mml:mi>\n                                <mml:mo>=<\/mml:mo>\n                                <mml:mn>1<\/mml:mn>\n                              <\/mml:mrow>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi>\n                                  \u2113\n                                  \n                                <\/mml:mi>\n                              <\/mml:mrow>\n                            <\/mml:munderover>\n                            <mml:msub>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi mathvariant=\"fraktur\">S<\/mml:mi>\n                              <\/mml:mrow>\n                              <mml:mi>k<\/mml:mi>\n                            <\/mml:msub>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">{\\mathbb P}^2(\\mathbb {K})\\setminus \\mathfrak {B}=\\bigcup _{k=1}^{\\ell } \\mathfrak {S}_k<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    such that, if\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"left-parenthesis German s 0 colon German t 0 colon German v 0 right-parenthesis element-of German upper S Subscript k\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:msub>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi mathvariant=\"fraktur\">s<\/mml:mi>\n                              <\/mml:mrow>\n                              <mml:mn>0<\/mml:mn>\n                            <\/mml:msub>\n                            <mml:mo>:<\/mml:mo>\n                            <mml:msub>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi mathvariant=\"fraktur\">t<\/mml:mi>\n                              <\/mml:mrow>\n                              <mml:mn>0<\/mml:mn>\n                            <\/mml:msub>\n                            <mml:mo>:<\/mml:mo>\n                            <mml:msub>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi mathvariant=\"fraktur\">v<\/mml:mi>\n                              <\/mml:mrow>\n                              <mml:mn>0<\/mml:mn>\n                            <\/mml:msub>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <mml:mo>\n                              \u2208\n                              \n                            <\/mml:mo>\n                            <mml:msub>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi mathvariant=\"fraktur\">S<\/mml:mi>\n                              <\/mml:mrow>\n                              <mml:mi>k<\/mml:mi>\n                            <\/mml:msub>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">(\\mathfrak {s}_0:\\mathfrak {t}_0:\\mathfrak {v}_0)\\in \\mathfrak {S}_k<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , then\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"script upper P left-parenthesis German s 0 comma German t 0 comma German v 0 right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">P<\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:msub>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi mathvariant=\"fraktur\">s<\/mml:mi>\n                              <\/mml:mrow>\n                              <mml:mn>0<\/mml:mn>\n                            <\/mml:msub>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:msub>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi mathvariant=\"fraktur\">t<\/mml:mi>\n                              <\/mml:mrow>\n                              <mml:mn>0<\/mml:mn>\n                            <\/mml:msub>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:msub>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi mathvariant=\"fraktur\">v<\/mml:mi>\n                              <\/mml:mrow>\n                              <mml:mn>0<\/mml:mn>\n                            <\/mml:msub>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\mathcal {P}(\\mathfrak {s}_0,\\mathfrak {t}_0,\\mathfrak {v}_0)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    is a point of\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"script upper S\">\n                        <mml:semantics>\n                          <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                            <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">S<\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\mathcal {S}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    of multiplicity\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"k\">\n                        <mml:semantics>\n                          <mml:mi>k<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">k<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    .\n                  <\/p>","DOI":"10.1090\/s0025-5718-2014-02907-4","type":"journal-article","created":{"date-parts":[[2014,10,9]],"date-time":"2014-10-09T15:10:16Z","timestamp":1412867416000},"page":"1991-2021","source":"Crossref","is-referenced-by-count":11,"title":["Computing the singularities of rational surfaces"],"prefix":"10.1090","volume":"84","author":[{"given":"S.","family":"P\u00e9rez-D\u00edaz","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"J.","family":"Sendra","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"C.","family":"Villarino","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[2014,10,9]]},"reference":[{"issue":"2","key":"1","doi-asserted-by":"publisher","first-page":"92","DOI":"10.1016\/j.jsc.2007.10.003","article-title":"Computing singular points of plane rational 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Rafael","year":"2008","ISBN":"https:\/\/id.crossref.org\/isbn\/9783540737247"},{"key":"13","isbn-type":"print","volume-title":"Basic algebraic geometry. 1","author":"Shafarevich, Igor R.","year":"1994","ISBN":"https:\/\/id.crossref.org\/isbn\/3540548122","edition":"2"},{"key":"14","series-title":"Texts and Monographs in Symbolic Computation","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/BF01191382","volume-title":"Polynomial algorithms in computer algebra","author":"Winkler, F.","year":"1996","ISBN":"https:\/\/id.crossref.org\/isbn\/3211827595"},{"key":"15","unstructured":"O. Zariski, P. Samuel, Commutative Algebra, Volume I. Graduate Texts in Mathematics, vol. 28, Springer-Verlag, 1975."}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/2015-84-294\/S0025-5718-2014-02907-4\/S0025-5718-2014-02907-4.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2015-84-294\/S0025-5718-2014-02907-4\/S0025-5718-2014-02907-4.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T18:24:34Z","timestamp":1776795874000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2015-84-294\/S0025-5718-2014-02907-4\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2014,10,9]]},"references-count":15,"journal-issue":{"issue":"294","published-print":{"date-parts":[[2015,7]]}},"alternative-id":["S0025-5718-2014-02907-4"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-2014-02907-4","archive":["CLOCKSS","Portico"],"relation":{},"ISSN":["1088-6842","0025-5718"],"issn-type":[{"value":"1088-6842","type":"electronic"},{"value":"0025-5718","type":"print"}],"subject":[],"published":{"date-parts":[[2014,10,9]]}}}