{"paper":{"title":"Birch's theorem with shifts","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Sam Chow","submitted_at":"2014-10-28T20:20:27Z","abstract_excerpt":"Let $f_1, ..., f_R$ be rational forms of degree $d \\ge 2$ in $n > \\sigma + R(R+1)(d-1)2^{d-1}$ variables, where $\\sigma$ is the dimension of the affine variety cut out by the condition $\\mathrm{rank}(\\nabla f_k)_{k=1}^R < R$. Assume that $\\mathbf{f} = \\mathbf{0}$ has a nonsingular real solution, and that the forms $(1,...,1) \\cdot \\nabla f_k$ are linearly independent. Let $\\boldsymbol{\\tau} \\in \\mathbb{R}^R$, let $\\mu$ be an irrational real number, and let $\\eta$ be a positive real number. We consider the values taken by $\\mathbf{f}(x_1 + \\mu, ..., x_n + \\mu)$ for integers $x_1, ..., x_n$. We "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1410.7789","kind":"arxiv","version":2},"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"}