{"paper":{"title":"On the equation $f(g(x)) =f(x)h^m(x)$ for composite polynomials","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Himadri Ganguli, Jonas Jankauskas","submitted_at":"2012-02-02T16:04:04Z","abstract_excerpt":"In this paper we solve the equation $f(g(x))=f(x)h^m(x)$ where $f(x)$, $g(x)$ and $h(x)$ are unknown polynomials with coefficients in an arbitrary field $K$, $f(x)$ is non-constant and separable, $\\deg g \\geq 2$, the polynomial $g(x)$ has non-zero derivative $g'(x) \\ne 0$ in $K[x]$ and the integer $m \\geq 2$ is not divisible by the characteristic of the field $K$. We prove that this equation has no solutions if $\\deg f \\geq 3$. If $\\deg f = 2$, we prove that $m = 2$ and give all solutions explicitly in terms of Chebyshev polynomials. The diophantine applications for such polynomials $f(x)$, $g"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1202.0471","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}