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It is known that the set <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$${\\mathcal {S}}(f)$$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>S<\/mml:mi>\n                    <mml:mo>(<\/mml:mo>\n                    <mml:mi>f<\/mml:mi>\n                    <mml:mo>)<\/mml:mo>\n                  <\/mml:mrow>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula> of points at which <jats:italic>f<\/jats:italic> is not finite forms an algebraic hypersurface in <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$${\\mathbb {K}}^n$$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msup>\n                    <mml:mrow>\n                      <mml:mi>K<\/mml:mi>\n                    <\/mml:mrow>\n                    <mml:mi>n<\/mml:mi>\n                  <\/mml:msup>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula>. The coordinate-wise valuation of <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$${\\mathcal {S}}(f)\\cap ({\\mathbb {K}}^*)^n$$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>S<\/mml:mi>\n                    <mml:mrow>\n                      <mml:mo>(<\/mml:mo>\n                      <mml:mi>f<\/mml:mi>\n                      <mml:mo>)<\/mml:mo>\n                    <\/mml:mrow>\n                    <mml:mo>\u2229<\/mml:mo>\n                    <mml:msup>\n                      <mml:mrow>\n                        <mml:mo>(<\/mml:mo>\n                        <mml:msup>\n                          <mml:mrow>\n                            <mml:mi>K<\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:mo>\u2217<\/mml:mo>\n                        <\/mml:msup>\n                        <mml:mo>)<\/mml:mo>\n                      <\/mml:mrow>\n                      <mml:mi>n<\/mml:mi>\n                    <\/mml:msup>\n                  <\/mml:mrow>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula> is a piecewise-linear object in <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$${\\mathbb {R}}^n$$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msup>\n                    <mml:mrow>\n                      <mml:mi>R<\/mml:mi>\n                    <\/mml:mrow>\n                    <mml:mi>n<\/mml:mi>\n                  <\/mml:msup>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula>, which we call the tropical non-properness set of <jats:italic>f<\/jats:italic>. We show that the tropical polynomial map corresponding to <jats:italic>f<\/jats:italic> has fibers satisfying a particular combinatorial degeneracy condition exactly over points in the tropical non-properness set of <jats:italic>f<\/jats:italic>. We then use this description to outline a polyhedral method for computing this set, and to recover the fan dual to the Newton polytope of the set at which a complex polynomial map is not finite. The proofs rely on classical correspondence and structural results from tropical geometry, combined with a new description of <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$${\\mathcal {S}}(f)$$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>S<\/mml:mi>\n                    <mml:mo>(<\/mml:mo>\n                    <mml:mi>f<\/mml:mi>\n                    <mml:mo>)<\/mml:mo>\n                  <\/mml:mrow>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula> in terms of multivariate resultants.<\/jats:p>","DOI":"10.1007\/s00454-024-00684-4","type":"journal-article","created":{"date-parts":[[2024,8,8]],"date-time":"2024-08-08T14:02:42Z","timestamp":1723125762000},"page":"1053-1078","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["The Tropical Non-Properness Set of a Polynomial Map"],"prefix":"10.1007","volume":"73","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-9654-906X","authenticated-orcid":false,"given":"Boulos","family":"El Hilany","sequence":"first","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2024,8,8]]},"reference":[{"issue":"3","key":"684_CR1","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/BF01075595","volume":"9","author":"DN Bernstein","year":"1975","unstructured":"Bernstein, D.N.: The number of roots of a system of equations. 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