Small Zeros of Quadratic Forms with Linear Conditions
نویسنده
چکیده
where H here stands for height of x and F , respectively. This generalizes a well known result of Cassels [2] about the existence of small zeros of quadratic forms with rational coefficients to the existence of small zeros of quadratic polynomials with rational coefficients. We generalize Masser’s result in the following way. Let K be a number field of degree d over Q. Let the coefficients fij be in K. Let M be a positive integer. Let L1(X), ..., LM (X) be linear forms in N + 1 variables with coefficients in K. Suppose there exists a point t ∈ K such that F (t) = 0, and Li(t) 6= 0 for each 1 ≤ i ≤ M . Then we prove that there exists such a point of bounded height. The bound on height is in terms of the heights of quadratic and linear forms, and reduces (up to a constant) to Masser’s type result over a number field in case M = 1 and L1(X) = X0.
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