On upper bounds of chalk and HUA for exponential sums

Let f be a polynomial of degree d with integer coefficients, p any prime, m any positive integer and S(f, p(m)) the exponential sum S(f,p(m)) = Sigmap (x=1)(m) e(p)(m) (f(x)). We establish that if f is nonconstant when read (m od p), then \S(f,p(m))\ less than or equal to 4:41p(m(1-1/d)). Let t = ord( p)(f')(1) let alpha be a zero of the congruence p(-t) f'(x) equivalent to 0 (mod p) of multiplicity v and let S-alpha(f,p(m)) be the sum S(f, p(m)) wi th x restricted to values congruent to alpha (mod p(m)). We obtain \S-alpha (f,p(m))\ less than or equal to min {v, 3:06}p(t/v +1)p(m(1-1/v+1)) for p o dd, m greater than or equal to t+2 and d(p)(f) greater than or equal to 1. If, in addition, p greater than or equal to (d-1)((2d)/(d-2)), then we obta in the sharp upper bound \S alpha (f,p(m))\ less than or equal to p(m(1-1/v +1)).