{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T05:43:11Z","timestamp":1776836591427,"version":"3.51.2"},"reference-count":18,"publisher":"American Mathematical Society (AMS)","issue":"340","license":[{"start":{"date-parts":[[2023,10,26]],"date-time":"2023-10-26T00:00:00Z","timestamp":1698278400000},"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                    We consider the tropicalization of tangent lines to a complete intersection curve\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper X\">\n                        <mml:semantics>\n                          <mml:mi>X<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">X<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    in\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"double-struck upper P Superscript n\">\n                        <mml:semantics>\n                          <mml:msup>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mi mathvariant=\"double-struck\">P<\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:mi>n<\/mml:mi>\n                          <\/mml:msup>\n                          <mml:annotation encoding=\"application\/x-tex\">\\mathbb {P}^n<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . Under mild hypotheses, we describe a procedure for computing the tropicalization of the image of the Gauss map of\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper X\">\n                        <mml:semantics>\n                          <mml:mi>X<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">X<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    in terms of the tropicalizations of the hypersurfaces cutting out\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper X\">\n                        <mml:semantics>\n                          <mml:mi>X<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">X<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . We apply this to obtain descriptions of the tropicalization of the dual variety\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper X Superscript asterisk\">\n                        <mml:semantics>\n                          <mml:msup>\n                            <mml:mi>X<\/mml:mi>\n                            <mml:mo>\n                              \u2217\n                              \n                            <\/mml:mo>\n                          <\/mml:msup>\n                          <mml:annotation encoding=\"application\/x-tex\">X^*<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    and tangential variety\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"tau left-parenthesis upper X right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>\n                              \u03c4\n                              \n                            <\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>X<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\tau (X)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    of\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper X\">\n                        <mml:semantics>\n                          <mml:mi>X<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">X<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . In particular, we are able to compute the degrees of\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper X Superscript asterisk\">\n                        <mml:semantics>\n                          <mml:msup>\n                            <mml:mi>X<\/mml:mi>\n                            <mml:mo>\n                              \u2217\n                              \n                            <\/mml:mo>\n                          <\/mml:msup>\n                          <mml:annotation encoding=\"application\/x-tex\">X^*<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    and\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"tau left-parenthesis upper X right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>\n                              \u03c4\n                              \n                            <\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>X<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\tau (X)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    and the Newton polytope of\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"tau left-parenthesis upper X right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>\n                              \u03c4\n                              \n                            <\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>X<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\tau (X)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    without using any elimination theory.\n                  <\/p>","DOI":"10.1090\/mcom\/3782","type":"journal-article","created":{"date-parts":[[2022,9,14]],"date-time":"2022-09-14T10:32:56Z","timestamp":1663151576000},"page":"931-979","source":"Crossref","is-referenced-by-count":0,"title":["Tropical tangents for complete intersection curves"],"prefix":"10.1090","volume":"92","author":[{"given":"Nathan","family":"Ilten","sequence":"first","affiliation":[]},{"given":"Yoav","family":"Len","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2022,10,26]]},"reference":[{"key":"1","doi-asserted-by":"publisher","first-page":"74","DOI":"10.5427\/jsing.2012.4e","article-title":"Inflection points of real and tropical plane curves","volume":"4","author":"Brugall\u00e9, Erwan A.","year":"2012","journal-title":"J. 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