On positive definite quadratic polynomials

R. Cook; S. Raghavan

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Let $Q_{1}, \dots, Q_{r}$ be quadratic forms with real coefficients. We prove that for any $ϵ > 0$ the system of inequalities $| Q_{1} (x) | < ϵ, \dots, | Q_{r} (x) | < ϵ$ has a nonzero integer solution, provided that the system $Q_{1} (x) = 0, \dots, Q_{r} (x) = 0$ has a nonsingular real solution and all forms in the real pencil generated by $Q_{1}, \dots, Q_{r}$ are irrational and have rank $> 8 r$ .

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