arXiv:1803.09306v4
Simultaneous Diophantine approximation: sums of squares and homogeneous polynomials
math.NTmath.DS11J1311J54
Abstract
Let be a homogeneous polynomial with rational coefficients in variables. We prove several results concerning uniform simultaneous approximation to points on the graph of , as well as on the hypersurface . The results are first stated for the case which is of particular interest.
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Detailed mathematical audit
01Statements2 reported findingsCorrect
The sharp Dirichlet-type bounds for simultaneous approximation subject to sums-of-squares constraints, including the general quadratic-form variants, are correct.
Theorems 2.1 and 2.2Correct
Simultaneous approximation with sums of squares
Section 2 · Theorems 2.1–2.2 · arXiv:1803.09306v4
The simplex lemma confines rational approximants of bounded height to a rational hyperplane. Combined with the transference inequalities for uniform exponents, the case analysis gives the asserted approximation exponent and shows sharpness in the stated dimension ranges.
Full paper, version 4 ↗Theorem 3.1 and statements 1a–3aCorrect
Quadratic-form generalizations
Section 3 and Sections 7–9 · Theorem 3.1 and statements 1a–3a · arXiv:1803.09306v4
Diagonalization and rational parametrization reduce each nonsingular quadratic constraint to the same lattice-exponent problem. The signature and isotropic-rank cases treated in Sections 7–9 exhaust the hypotheses and give the displayed bounds.
02Proofs2 reported findingsCorrect
The simplex lemma, transference step, and quadratic-form case analysis are correct and complete.
Sections 4–6Correct and complete
Simplex and transference estimates have compatible constants
Sections 4–6 · arXiv:1803.09306v4
The determinant lower bound for distinct rational points contradicts the upper bound supplied by a small approximation ball unless the points lie on one rational hyperplane. The Marnat–Moshchevitin transference inequality then converts the resulting dual obstruction to the claimed simultaneous exponent with the stated endpoints.
Sections 7–9Correct and complete
All quadratic signatures are covered
Sections 7–9 · arXiv:1803.09306v4
The proof separates anisotropic, split, and intermediate-rank forms. In each case the rational linear change of coordinates preserves exponents up to fixed constants, and the parametrization or hyperplane argument supplies both the upper and lower bounds required by the corresponding statement.
03Novelty0 reported findingsNo non-novelty findings
No non-novelty findings.