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Comp."],"abstract":"<p>\n                    Algorithms working with linear algebraic groups often represent them via defining polynomial equations. One can always choose defining equations for an algebraic group to be of degree at most the degree of the group as an algebraic variety. However, the degree of a linear algebraic group\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper G subset-of upper G upper L Subscript n Baseline left-parenthesis upper C right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>G<\/mml:mi>\n                            <mml:mo>\n                              \u2282\n                              \n                            <\/mml:mo>\n                            <mml:msub>\n                              <mml:mi>GL<\/mml:mi>\n                              <mml:mi>n<\/mml:mi>\n                            <\/mml:msub>\n                            <mml:mo>\n                              \u2061\n                              \n                            <\/mml:mo>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>C<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">G \\subset \\operatorname {GL}_n(C)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    can be arbitrarily large even for\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"n equals 1\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>n<\/mml:mi>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">n = 1<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . One of the key ingredients of Hrushovski\u2019s algorithm for computing the Galois group of a linear differential equation was an idea to \u201capproximate\u201d every algebraic subgroup of\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper G upper L Subscript n Baseline left-parenthesis upper C right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:msub>\n                              <mml:mi>GL<\/mml:mi>\n                              <mml:mi>n<\/mml:mi>\n                            <\/mml:msub>\n                            <mml:mo>\n                              \u2061\n                              \n                            <\/mml:mo>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>C<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\operatorname {GL}_n(C)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    by a \u201csimilar\u201d group so that the degree of the latter is bounded uniformly in\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"n\">\n                        <mml:semantics>\n                          <mml:mi>n<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">n<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . Making this uniform bound computationally feasible is crucial for making the algorithm practical.\n                  <\/p>\n                  <p>\n                    In this paper, we derive a single-exponential degree bound for such an approximation (we call it a\n                    <italic>toric envelope<\/italic>\n                    ), which is qualitatively optimal. As an application, we improve the quintuply exponential bound due to Feng for the first step of Hrushovski\u2019s algorithm to a single-exponential bound. For the cases\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"n equals 2 comma 3\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>n<\/mml:mi>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mn>2<\/mml:mn>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mn>3<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">n = 2, 3<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    often arising in practice, we further refine our general bound.\n                  <\/p>","DOI":"10.1090\/mcom\/3695","type":"journal-article","created":{"date-parts":[[2021,10,27]],"date-time":"2021-10-27T09:40:49Z","timestamp":1635327649000},"page":"1501-1519","source":"Crossref","is-referenced-by-count":3,"title":["Degree bound for toric envelope of a linear algebraic group"],"prefix":"10.1090","volume":"91","author":[{"given":"Eli","family":"Amzallag","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Andrei","family":"Minchenko","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Gleb","family":"Pogudin","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[2021,10,28]]},"reference":[{"key":"1","isbn-type":"print","doi-asserted-by":"publisher","first-page":"63","DOI":"10.1145\/2930889.2930932","article-title":"Computing the Lie algebra of the differential Galois group of a linear differential system","author":"Barkatou, Moulay","year":"2016","ISBN":"https:\/\/id.crossref.org\/isbn\/9781450343800"},{"key":"2","isbn-type":"print","first-page":"413","article-title":"Differential Galois theory","author":"Beukers, Frits","year":"1992","ISBN":"https:\/\/id.crossref.org\/isbn\/3540533427"},{"key":"3","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4612-0941-6","volume-title":"Linear algebraic groups","author":"Borel, Armand","year":"1991","edition":"2"},{"issue":"2","key":"4","first-page":"397","article-title":"Groupe de Picard et nombres caract\u00e9ristiques des vari\u00e9t\u00e9s sph\u00e9riques","volume":"58","author":"Brion, Michel","year":"1989","journal-title":"Duke Math. 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