Counting Problems Related To Intersecting Polynomials In Function Domain

Let N be the set of all natural numbers,write N+=N\{0},and let Fq be a finite field of q elements,whose the characteristic is p.Let A=Fq[t]be the polynomial ring over the Fq.Ω={ω∈ A:ω is a monic,irreducible polynomial}.For M E Z,let M=qM,GM={m∈A:deg m<M}.Let A(?)A.We say that A is a positive upper of density set,if Let f∈A[x].We call f to be intersective,if(A-A)∩I(f)≠(?),for any positive upper density subset of A(?)A,where I(f)=f(A)\{0}.Define counting function,which suffices the nunber of solutions of equation x-y=z with x,y∈A and z∈I(f).In 2013,Le and Liu first obtained counting estimates of intersective polynomial x2 over the field IFq.Let N E N+,p>2.Let A (?)GN with |A|=δN.If then for N sufficiently large,depending only on q,we have where C,C’≥ lare constants depending only on q.The second purpose of this paper to prove the following counting estimates of general intersective polynomials over A.Let ∈ N+,and h E A[x]be an intersective polynomial of degree k.Let A (?)GN with|A|=δN.Suppose that 2<k<p.If then for N sufficiently large,depending only on q and h,we have where C,C’≥1 are constants depending only on q and h.Since A(?)GN,it is easy to see thatWhen |A|>>N N,we find that Therefore,the exponent of N(1-1/k)is the best possibile.Let k=2,if then when N ≥e2,we have then,therefore,when k=2,the above result improve the estimate for x2 of Le and Liu.In 2011,Le and Spencer first investigated the corresponding version of theorem of Sarkozy in the setting of A.They published the following result.Let N E N+,u∈IFq\{0} and A (?) GN with |A|=δN.If(A-A)n(Ω+u)=(?),then where C>1 and 0<c<1 are constants depending only on q.Lin Yurong,Li Guoquan and the author have shown that there is a bug in their iteration argument.Also,by following the approach of Le and Spencer they proved the following result.Let N E N+,u∈Fq\{0}.Let A (?) GN with |A|=δN.If(A-A)n(Ω+u)=(?),thenδ≤Cq-cN1/3,where C>1 and 0<c<1 are constants depending only on q.As an direct consequence of the above conclusion,we see that ifδ≥Cq-cN1/3,then(A-A)n(Ω+u)≠(?).The first purpose of this paper to provide a quantitative version of the above statement.First,we give a weighted counting estimate for the condinality of the set(A-A)∩(Ω+u).Let N∈N+,A(?)GN with |A|=δN.Ifδ≥C1e-c1N1/3,then for N sufficiently large,depending only on q,we have where C1,C2,C3≥1 and 0<c1<1 are constants depending only on q,andInspecting the proof of the above result,we find thatδ≥Cq-cN1/3,is the best esimate that can be obtained by using the methods of Lê and Spencer.Furthermore,this result implies our quantitative version related to the set(A-A)∩(Ω+u).Let N ∈N+,A(?)GN with |A|=δN.Ifδ≥Cq-cN1/3,then for N sufficiently large,depending only on q,we have where C1,C2,C3≥1 and 0<c1<1 are constants depending only on q.