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230 7. Approximations and Inequalities

5.6. Begin by a careful study of the quadratic case. What happens if all

the xi have the same sign? Use an induction argument. The case that

zi 5 a < 0 < b 5 x,, can be disposed of by looking at the values

of the polynomial at a and b after dividing by (x - x1)(2: - x,,). As

for the case that a 5 xi 5 x, 5 b, note that p(a)p(b) is the product

of quadratics (a - xi)(b - zi) in the xi. The remaining cases can be

handled by similar types of consideration.

5.7. It has to be shown that G(X) = II(zj - zi)/(j - i) takes integer values

whenever the components of x are integers. Argue that it is enough

to consider nonnegative integer values and prove by induction on the

maximum of the zi. The result is clear if this maximum is /c. Now

fix h of the k + 1 variables and consider G as a function of a single

variable. Use the fact that a polynomial of degree not exceeding k

over Z which assumes integer values for at least k + 1 consecutive

integers assumes integer values at each integer point.

5.8. There is an obvious example of a polynomial with integer coefficients

which assumes arbitrarily small values on any open subinterval of

[-l,l]. Adapt this.

5.9. Draw a diagram.

5.10. Express this polynomial using Lagrange’s formula. What are the zeros
of the Tchebychev polynomials? Use this fact in your expression for
p(z). Look at the special case in which each yk is 1.