Some Examples And Problems Related To Intersecting Polynomials In Polynomial Rings Over Finite Field

Let N denote the set of all the natural numbers,Note N₊=N\{0}.Let A（?）N₊,The upper density of A is（?）.Where|A∩{1,2,...,N}|denote the cardinality of A∩{1,2,...,N}.Let F_qdenote the finite field of q elements,let A=F_q[t]be the polynomial ring over F_q.Let（?）={ω∈A:ωis a monic irreducible polynomial}.For N∈N₊,letG_N={m∈A:deg m<N}.In 2011,L?e and Spencer got the following result:Let N∈N₊,A（?）G_Nand r∈F_q\{0}.If（A-A）∩（（?）+r）=?,then|A|≤Cq^Nexp（-c N/log N）.Where C≥1 and 0<c<1 is a constant depending only on q.But,there is a loophole in their article.This article corrects this error and proves the following result:Let N∈N₊,A（?）G_Nand r∈F_q\{0}.If（A-A）∩（（?）+r）=?,then|A|≤Cq^Nexp(-?CN^1/3).Where C≥1 and?C>0 is a constant depending only on q.The second purpose of this paper is to prove（x²-13）（x²-17）（x²-221）,（x³-19）（x²+x+1）is the polnomial of the second kind,we can find a method which can judge form of（x²-n₁）（x²-n₂）（x²-n₃）is polynomial of the second kind.We obtain the following result:Let p₃is prime number in form of 4k+1,p₄is prime number in form of 8k+1,If they meet one of the following conditions:（a）there is b₂,meet p₃|b₂²-p₄;（b）there is b₃,meet p₄|b₃²-p₃;It can conclude（x²-p₃）（x²-p₄）（x²-p₃p₄）is polynomial of the second kind.