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This polynomial is known to have nonnegative and symmetric coefficients and is conjectured to be <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$$\\gamma$$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>\u03b3<\/mml:mi>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula>-positive when <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$$\\Delta$$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>\u0394<\/mml:mi>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula> is flag. This paper shows that the local <jats:italic>h<\/jats:italic>-polynomial has the stronger property of being real-rooted when <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$$\\Delta$$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>\u0394<\/mml:mi>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula> is the barycentric subdivision of an arbitrary geometric triangulation <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$$\\Gamma$$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>\u0393<\/mml:mi>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula> of the simplex. An analogous result for edgewise subdivisions is proven. The proofs are based on a new combinatorial formula for the local <jats:italic>h<\/jats:italic>-polynomial of <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$$\\Delta$$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>\u0394<\/mml:mi>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula>, which is valid when <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$$\\Delta$$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>\u0394<\/mml:mi>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula> is any uniform triangulation of <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$$\\Gamma$$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>\u0393<\/mml:mi>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula>. A combinatorial interpretation of the local <jats:italic>h<\/jats:italic>-polynomial of the second barycentric subdivision of the simplex is deduced.<\/jats:p>","DOI":"10.1007\/s00493-025-00162-2","type":"journal-article","created":{"date-parts":[[2025,6,23]],"date-time":"2025-06-23T03:37:57Z","timestamp":1750649877000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Local h-polynomials, Uniform Triangulations and Real-rootedness"],"prefix":"10.1007","volume":"45","author":[{"given":"Christos A.","family":"Athanasiadis","sequence":"first","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2025,6,23]]},"reference":[{"key":"162_CR1","doi-asserted-by":"crossref","unstructured":"Athanasiadis, C.A.: Flag subdivisions and $$\\gamma$$-vectors. Pacific J. 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