MathDB

39

Part of PEN E Problems

Problems(1)

E 39

Source:

1/20/2013
Let cc be a nonzero real number. Suppose that g(x)=c0xr+c1xr−1+⋯+cr−1x+crg(x)=c_0x^r+c_1x^{r-1}+\cdots+c_{r-1}x+c_r is a polynomial with integer coefficients. Suppose that the roots of g(x)g(x) are b1,⋯ ,brb_1,\cdots,b_r. Let kk be a given positive integer. Show that there is a prime pp such that p>max⁡(k,∣c∣,∣cr∣)p>\max(k,|c|,|c_r|), and moreover if tt is a real number between 00 and 11, and jj is one of 1,⋯ ,r1,\cdots,r, then ∣( cr bjg(tbj) )pe(1−t)b∣<(p−1)!2r.|(\text{ }c^r\text{ }b_j\text{}g(tb_j)\text{ })^pe^{(1-t)b}|<\dfrac{(p-1)!}{2r}. Furthermore, if f(x)=erp−1xp−1(g(x))p(p−1)!f(x)=\dfrac{e^{rp-1}x^{p-1}(g(x))^p}{(p-1)!} then ∣∑j=1r∫01e(1−t)bjf(tbj)dt∣≤12.\left|\sum_{j=1}^r\int_0^1 e^{(1-t)b_j}f(tb_j)dt\right|\leq \dfrac{1}{2}.
algebrapolynomialintegration