{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T04:43:32Z","timestamp":1776833012689,"version":"3.51.2"},"reference-count":20,"publisher":"American Mathematical Society (AMS)","issue":"325","license":[{"start":{"date-parts":[[2021,2,20]],"date-time":"2021-02-20T00:00:00Z","timestamp":1613779200000},"content-version":"am","delay-in-days":366,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"funder":[{"DOI":"10.13039\/501100003246","name":"Nederlandse Organisatie voor Wetenschappelijk Onderzoek","doi-asserted-by":"publisher","award":["Vici 639.033.514 (Jan Draisma)"],"award-info":[{"award-number":["Vici 639.033.514 (Jan Draisma)"]}],"id":[{"id":"10.13039\/501100003246","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100003246","name":"Nederlandse Organisatie voor Wetenschappelijk Onderzoek","doi-asserted-by":"publisher","award":["MDM-2014-0445"],"award-info":[{"award-number":["MDM-2014-0445"]}],"id":[{"id":"10.13039\/501100003246","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100003329","name":"Ministerio de Econom\u00c3\u00ada y Competitividad","doi-asserted-by":"publisher","award":["Vici 639.033.514 (Jan Draisma)"],"award-info":[{"award-number":["Vici 639.033.514 (Jan Draisma)"]}],"id":[{"id":"10.13039\/501100003329","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100003329","name":"Ministerio de Econom\u00c3\u00ada y Competitividad","doi-asserted-by":"publisher","award":["MDM-2014-0445"],"award-info":[{"award-number":["MDM-2014-0445"]}],"id":[{"id":"10.13039\/501100003329","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100005156","name":"Alexander von Humboldt-Stiftung","doi-asserted-by":"publisher","award":["Vici 639.033.514 (Jan Draisma)"],"award-info":[{"award-number":["Vici 639.033.514 (Jan Draisma)"]}],"id":[{"id":"10.13039\/100005156","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100005156","name":"Alexander von Humboldt-Stiftung","doi-asserted-by":"publisher","award":["MDM-2014-0445"],"award-info":[{"award-number":["MDM-2014-0445"]}],"id":[{"id":"10.13039\/100005156","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    We introduce the\n                    <italic>monic rank<\/italic>\n                    of a vector relative to an affine-hyperplane section of an irreducible Zariski-closed affine cone\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 show that the monic rank is finite and greater than or equal to the usual\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                    -rank. We describe an algorithmic technique based on classical invariant theory to determine, in concrete situations, the maximal monic rank. Using this technique, we establish three new instances of a conjecture due to B.\u00a0Shapiro which states that a binary form of degree\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"d dot e\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>d<\/mml:mi>\n                            <mml:mo>\n                              \u22c5\n                              \n                            <\/mml:mo>\n                            <mml:mi>e<\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">d\\cdot e<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    is the sum of\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"d\">\n                        <mml:semantics>\n                          <mml:mi>d<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">d<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"d\">\n                        <mml:semantics>\n                          <mml:mi>d<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">d<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    th powers of forms of degree\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"e\">\n                        <mml:semantics>\n                          <mml:mi>e<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">e<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . Furthermore, in the case where\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                    is the cone of highest weight vectors in an irreducible representation\u2014this includes the well-known cases of tensor rank and symmetric rank\u2014we raise the question whether the maximal rank\n                    <italic>equals<\/italic>\n                    the maximal monic rank. We answer this question affirmatively in several instances.\n                  <\/p>","DOI":"10.1090\/mcom\/3512","type":"journal-article","created":{"date-parts":[[2019,11,27]],"date-time":"2019-11-27T10:02:17Z","timestamp":1574848937000},"page":"2481-2505","source":"Crossref","is-referenced-by-count":0,"title":["The monic rank"],"prefix":"10.1090","volume":"89","author":[{"given":"Arthur","family":"Bik","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jan","family":"Draisma","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Alessandro","family":"Oneto","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Emanuele","family":"Ventura","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[2020,2,20]]},"reference":[{"issue":"2","key":"1","first-page":"201","article-title":"Polynomial interpolation in several variables","volume":"4","author":"Alexander, J.","year":"1995","journal-title":"J. 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