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      發布時間:2020-07-02文章來源:劉麗 瀏覽次數:

      報告題目:  Subset sums over Galois rings

      人:周海燕 (南京師范大學)

         間:202076日 上午 9:00-10:00





      Let 1\leq k\leq n be a positive integer and G be a finitely generated abelian group. Let D=\{a_1, a_2, \cdots, a_n\}\subset G, b\in G$. Let $N(k,b) denote the number of nonempty subsets with length k which sums to b, i.e., N(k,b)=N_D(k,b)=\#\{S\subsetD|\sum_{a\inS}a=b,\|S|=k\}=\frac{1}{k!}\#\{x_1+x_2+\cdots+x_k=b|x_i\in D,\ distinct\}.The decision version of the k-subset sum problem for D is to determine if N_D(k, b)>0.  It arises from several applications in coding theory, cryptography, graph theory and some other fields.  From a mathematical point of view, we are more interested in the actual value of the solution number N_D(k,b). We would like to have an explicit formula or an asymptotic formula for the solution number N_D(k,b), but this is apparently too much to hope for in general. However, we believe that it should be possible to obtain an asymptotic formula or the number N_D(k,b) for k in a certain range if D is close to a large subset of G with certain algebraic structure.  For example, let \mathbb{F}_q be the finite field with q=p^s elements, where p is a prime and s\geq 1 is an integer,  it is known that if \mathbb{F}_{q}-D is a small set, there is a simple asymptotic formula for N_D(k,b). We are interested in the subset sums problem over a Galois ring R=GR(p^2,p^{2r}). Let R(k)=\{y\in R|~y=x^k~ {\rm for~some}~x\in R\}. We obtain the asymptotic formula of N_{D}(n,b) for D=R(k)\setminus\{0\}=R(k)^\ast. From which we deduce that any element in R is the sum of n different k-th power under sufficient condition.

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