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\n We show that the number\n \n \n \n \n \n \u03b1\n \n <\/mml:mi>\n =<\/mml:mo>\n (<\/mml:mo>\n 1<\/mml:mn>\n +<\/mml:mo>\n \n 3<\/mml:mn>\n +<\/mml:mo>\n 2<\/mml:mn>\n \n 5<\/mml:mn>\n <\/mml:msqrt>\n <\/mml:msqrt>\n )<\/mml:mo>\n \n \/<\/mml:mo>\n <\/mml:mrow>\n 2<\/mml:mn>\n <\/mml:mrow>\n \\alpha =(1+\\sqrt {3+2\\sqrt {5}})\/2<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n with minimal polynomial\n \n \n \n \n \n x<\/mml:mi>\n 4<\/mml:mn>\n <\/mml:msup>\n \n \u2212\n \n <\/mml:mo>\n 2<\/mml:mn>\n \n x<\/mml:mi>\n 3<\/mml:mn>\n <\/mml:msup>\n +<\/mml:mo>\n x<\/mml:mi>\n \n \u2212\n \n <\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n x^4-2x^3+x-1<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n is the only Pisot number whose four distinct conjugates\n \n \n \n \n \n \n \u03b1\n \n <\/mml:mi>\n 1<\/mml:mn>\n <\/mml:msub>\n ,<\/mml:mo>\n \n \n \u03b1\n \n <\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msub>\n ,<\/mml:mo>\n \n \n \u03b1\n \n <\/mml:mi>\n 3<\/mml:mn>\n <\/mml:msub>\n ,<\/mml:mo>\n \n \n \u03b1\n \n <\/mml:mi>\n 4<\/mml:mn>\n <\/mml:msub>\n <\/mml:mrow>\n \\alpha _1,\\alpha _2,\\alpha _3,\\alpha _4<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n satisfy the additive relation\n \n \n \n \n \n \n \u03b1\n \n <\/mml:mi>\n 1<\/mml:mn>\n <\/mml:msub>\n +<\/mml:mo>\n \n \n \u03b1\n \n <\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msub>\n =<\/mml:mo>\n \n \n \u03b1\n \n <\/mml:mi>\n 3<\/mml:mn>\n <\/mml:msub>\n +<\/mml:mo>\n \n \n \u03b1\n \n <\/mml:mi>\n 4<\/mml:mn>\n <\/mml:msub>\n <\/mml:mrow>\n \\alpha _1+\\alpha _2=\\alpha _3+\\alpha _4<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . This implies that there exists no two non-real conjugates of a Pisot number with the same imaginary part and also that at most two conjugates of a Pisot number can have the same real part. On the other hand, we prove that similar four term equations\n \n \n \n \n \n \n \u03b1\n \n <\/mml:mi>\n 1<\/mml:mn>\n <\/mml:msub>\n =<\/mml:mo>\n \n \n \u03b1\n \n <\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msub>\n +<\/mml:mo>\n \n \n \u03b1\n \n <\/mml:mi>\n 3<\/mml:mn>\n <\/mml:msub>\n +<\/mml:mo>\n \n \n \u03b1\n \n <\/mml:mi>\n 4<\/mml:mn>\n <\/mml:msub>\n <\/mml:mrow>\n \\alpha _1 = \\alpha _2 + \\alpha _3+\\alpha _4<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n or\n \n \n \n \n \n \n \u03b1\n \n <\/mml:mi>\n 1<\/mml:mn>\n <\/mml:msub>\n +<\/mml:mo>\n \n \n \u03b1\n \n <\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msub>\n +<\/mml:mo>\n \n \n \u03b1\n \n <\/mml:mi>\n 3<\/mml:mn>\n <\/mml:msub>\n +<\/mml:mo>\n \n \n \u03b1\n \n <\/mml:mi>\n 4<\/mml:mn>\n <\/mml:msub>\n =<\/mml:mo>\n 0<\/mml:mn>\n <\/mml:mrow>\n \\alpha _1 + \\alpha _2 + \\alpha _3 + \\alpha _4 =0<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n cannot be solved in conjugates of a Pisot number\n \n \n \n \n \u03b1\n \n <\/mml:mi>\n \\alpha<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . We also show that the roots of the Siegel\u2019s polynomial\n \n \n \n \n \n x<\/mml:mi>\n 3<\/mml:mn>\n <\/mml:msup>\n \n \u2212\n \n <\/mml:mo>\n x<\/mml:mi>\n \n \u2212\n \n <\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n x^3-x-1<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n are the only solutions to the three term equation\n \n \n \n \n \n \n \u03b1\n \n <\/mml:mi>\n 1<\/mml:mn>\n <\/mml:msub>\n +<\/mml:mo>\n \n \n \u03b1\n \n <\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msub>\n +<\/mml:mo>\n \n \n \u03b1\n \n <\/mml:mi>\n 3<\/mml:mn>\n <\/mml:msub>\n =<\/mml:mo>\n 0<\/mml:mn>\n <\/mml:mrow>\n \\alpha _1+\\alpha _2+\\alpha _3=0<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n in conjugates of a Pisot number. Finally, we prove that there exists no Pisot number whose conjugates satisfy the relation\n \n \n \n \n \n \n \u03b1\n \n <\/mml:mi>\n 1<\/mml:mn>\n <\/mml:msub>\n =<\/mml:mo>\n \n \n \u03b1\n \n <\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msub>\n +<\/mml:mo>\n \n \n \u03b1\n \n <\/mml:mi>\n 3<\/mml:mn>\n <\/mml:msub>\n <\/mml:mrow>\n \\alpha _1=\\alpha _2+\\alpha _3<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n .\n <\/p>","DOI":"10.1090\/mcom\/3103","type":"journal-article","created":{"date-parts":[[2015,9,2]],"date-time":"2015-09-02T12:42:03Z","timestamp":1441197723000},"page":"935-950","source":"Crossref","is-referenced-by-count":5,"title":["No two non-real conjugates of a Pisot number have the same imaginary part"],"prefix":"10.1090","volume":"86","author":[{"given":"Art\u016bras","family":"Dubickas","sequence":"first","affiliation":[]},{"given":"Kevin","family":"Hare","sequence":"additional","affiliation":[]},{"given":"Jonas","family":"Jankauskas","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2016,4,13]]},"reference":[{"key":"1","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-0348-8632-1","volume-title":"Pisot and Salem numbers","author":"Bertin, M.-J.","year":"1992","ISBN":"https:\/\/id.crossref.org\/isbn\/3764326484"},{"issue":"2","key":"2","doi-asserted-by":"publisher","first-page":"103","DOI":"10.4064\/aa-79-2-103-111","article-title":"Lower bounds of heights of points on hypersurfaces","volume":"79","author":"Beukers, Frits","year":"1997","journal-title":"Acta Arith.","ISSN":"https:\/\/id.crossref.org\/issn\/0065-1036","issn-type":"print"},{"issue":"144","key":"3","doi-asserted-by":"publisher","first-page":"1244","DOI":"10.2307\/2006349","article-title":"Pisot and Salem numbers in intervals of the real line","volume":"32","author":"Boyd, David W.","year":"1978","journal-title":"Math. 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