To any combinatorial triangulation T of a square, there is an associated polynomial relation pT\documentclass[12pt]{minimal}
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\begin{document}$$p_T$$\end{document} among the areas of the triangles of T. With the goal of understanding this polynomial, we consider polynomials obtained from pT\documentclass[12pt]{minimal}
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\begin{document}$$p_T$$\end{document} by choosing l of its variables and specializing pT\documentclass[12pt]{minimal}
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\begin{document}$$p_T$$\end{document} to these variables by zeroing out the remaining variables. We show that for fixed l, the set [inline-graphic not available: see fulltext] of integer polynomials that appear as irreducible factors of such specializations is finite. We compute this area encyclopedia[inline-graphic not available: see fulltext] for l⩽4\documentclass[12pt]{minimal}
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\begin{document}$$l\leqslant 4$$\end{document}. We also show that in any dissection of a square into l triangles, the areas of the triangles must satisfy a polynomial in [inline-graphic not available: see fulltext]. Our results are obtained by studying the rational map that associates to each drawing of T the tuple of areas of the triangles in that drawing. By analyzing the ways of approaching the base locus, we derive restrictions on points of the closure of the image of this map.
