{"paper":{"title":"Low rank approximation of polynomials","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AC"],"primary_cat":"math.CO","authors_text":"Alexander Schrijver","submitted_at":"2012-11-15T10:46:47Z","abstract_excerpt":"Let $k\\leq n$. Each polynomial $p\\in\\oR[x_1,...,x_n]$ can be uniquely written as $p=\\sum_{\\mu}\\mu p_{\\mu}$, where $\\mu$ ranges over the set $M$ of all monomials in $\\oR[x_1,...,x_k]$ and where $p_{\\mu}\\in\\oR[x_{k+1},...,x_n]$. If $p$ is $d$-homogeneous and $\\varepsilon>0$, we say that $p$ is {\\em $\\varepsilon$-concentrated on the first $k$ variables} if $$\\sum_{\\mu\\in M\\atop\\deg(\\mu)<d}\\max_{x\\in\\oR^{n-k}\\atop\\|x\\|=1}p_{\\mu}(x)^2\\leq\\varepsilon\\|p\\|^2,$$ where $\\|p\\|$ is the Bombieri norm of $p$. We show that for each $d\\in\\oN$ and $\\varepsilon>0$ there exists $k_{d,\\varepsilon}$ such that for"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1211.3569","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"}