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Comp."],"abstract":"<p>\n                    We consider the problem of computing homogeneous coordinates of points in a zero-dimensional subscheme of a compact, complex toric variety\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                    . Our starting point is a homogeneous ideal\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper I\">\n                        <mml:semantics>\n                          <mml:mi>I<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">I<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    in the Cox ring 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                    , which in practice might arise from homogenizing a sparse polynomial system. We prove a new eigenvalue theorem in the toric compact setting, which leads to a novel, robust numerical approach for solving this problem. Our method works in particular for systems having isolated solutions with arbitrary multiplicities. It depends on the multigraded regularity properties of\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper I\">\n                        <mml:semantics>\n                          <mml:mi>I<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">I<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . We study these properties and provide bounds on the size of the matrices in our approach when\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper I\">\n                        <mml:semantics>\n                          <mml:mi>I<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">I<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    is a complete intersection.\n                  <\/p>","DOI":"10.1090\/mcom\/3744","type":"journal-article","created":{"date-parts":[[2022,3,23]],"date-time":"2022-03-23T10:01:26Z","timestamp":1648029686000},"page":"2397-2429","source":"Crossref","is-referenced-by-count":7,"title":["Toric eigenvalue methods for solving sparse polynomial systems"],"prefix":"10.1090","volume":"91","author":[{"given":"Mat\u00edas","family":"Bender","sequence":"first","affiliation":[]},{"given":"Simon","family":"Telen","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2022,6,30]]},"reference":[{"issue":"4","key":"1","doi-asserted-by":"publisher","first-page":"1147","DOI":"10.4310\/PAMQ.2020.v16.n4.a12","article-title":"Immaculate line bundles on toric varieties","volume":"16","author":"Altmann, Klaus","year":"2020","journal-title":"Pure Appl. 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