Indefinite Quadratic Form

An indefinite quadratic form is a quadratic form that takes both positive and negative values. The quadratic form, written in the form , is indefinite if eigenvalues of the matrix are of both signs.

For an indefinite quadratic form with a real symmetric matrix having one positive and four negative eigenvalues, write . Kahlon (2026, Theorem 1.1) reports that every vector admits a lattice point satisfying

The constant 8 is sharp, giving for positive inhomogeneous values in this quadratic form signature. The proof combines analytic reduction with exact covering certificates.