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Given integers and a constant , consider the space of -tuples of real polynomials in variables of degree , whose coefficients are in absolute value, and satisfying . We study the family of algebraic functions, where is a polynomial, and being a constant depending only on . The main result is a quantitative extension theorem for these functions which is uniform in . This is used to prove Bernstein-type inequalities which are again uniform with respect to .
The proof...
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