Subset Sum On Galois Ring And Polynomial On Finite Field

In this paper,we study the subset sum problem over Galois rings and counting polynomials with a given number of zeros in finite fields.Let R be a Galois ring and D(?)R a finite subset.For a positive integer 1 ?K? |D| and an element b ?R,N(k,b)denotes the number of k-elements subsets S C D such that?x?sx=b In chapter 2,employing the sieve formula together with the Mobius Inversion For-mula,we obtain an explicit formula for N(k,b)when D=R.Let Fq[x]denote a polynomial ring in an indeterminate x over a finite field Fq with q=pm elements,where p is a prime and m is a positive integer.Let l,n(l<n)be two positive integers,vl(x)= alxl + al-1xn-1+…+a1x+ a0?Fq[x]of degree l and fix a monic polynomial u(x)= xn + un-1xn-1+…+ul+1xl+1?Fq[x]of degree n.For any non-negative integer k<n,let Nk(u(x),l)denote the total number of vl(x)such that u(x)+ vl(x)has exactly k roots in Fq.In chapter 3,we show the exact formulae about Nk(u(x),l)when n-l=1,2,the estimation of Nk(u(x),l)when n-l = k + 1 and the average number of zeros and corresponding variance for a polynomial when n-l= 2.