Diagonal Cubic Form,Periodic Function And Gawron-Miska-Ulas Conjecture

In this thesis,we mainly study three problems in number theory:counting the zeros of diagonal cubic form over finite fields,seeking for the formula of the smallest positive period of polynomial over residue classes ring and the number of polynomial functions with given smallest positive period over residue classes ring,and the Gawron-Miska-Ulas conjecture about the arithmetic properties of the coefficients of power series expansion of Π_n=0^∞(1-x²ⁿ)^m.Finding the formula for the number of zeros of diagonal equation is a difficult problem in number theory.Let p be a prime and let k be a positive integer.Denote by F_q the finite field with q=pk elements.Let s be a positive integer and a,y∈F_q.Let Ns（a）and Ts（y）stand for the number of zeros of the diagonal cubic forms x₁³+...+x_s³=a and x₁³+...x_s³+yxs+13=0 over F_q,respectively.When q=p≡1（mod 3）and y is a non-cubic element over F_p,Gauss proved that T₂（y）=p²+1/2（p-1）（-c+9d）,where c and is uniquely determined by 4p=c²+27d² and c≡1（mod 3）（except for the sign of d）.Chowla,Cowles and Cowles（1977）and Myerson（1979）proved that ∑_s=1^∞ N_s（0）x^s is a rational function of x and presented its explicit expression over F_p and F_q,respectively.When 2 is a cubic element in F_p,Chowla,Cowles and Cowles determined in 1978 the sign of d.But when 2 is a cubic element in F_p,the sign of d is still undecided.In Chapter 1,we solve the Gauss sign problem by determining the sign of d.Furthermore,we show that ∑_s=1^∞ N_s（α）x^s and ∑_s=1^∞ T_s（y）x^s are both rational functions of x and also supply their explicit expressions by using the method of exponential sums over finite fields.The second topic of this thesis is about the polynomial over residue class ring.If a map over a ring R can be induced by a polynomial over R,then we call this map a polynomial function.In 1964,Carlitz gave a necessary and sufficient condition for a map to be a polynomial function.Let p be a prime and let n be a positive integer.In 1967,Keller and Olson presented a formula for the number of polynomial functions over Z/pnZ.Let m be a positive integer and let f（x）∈ Z[x].If there is a positive integer T such that f（a+T）≡ f（a）（mod m）holds for all a ∈ Z,then we say T is a period of f（x）modulo m.As we know,the congruence f（a+m）≡f（a）（mod m）holds for all a ∈Z.That is,every polynomial f（x）over a residue class ring is periodic.One will naturally ask the following two questions:（1）.What is the smallest positive period（?）_m（f）of f modulo m?（2）.What is the number of polynomial functions with a given smallest positive period?In Chapter 2,we completely answer these problems.Actually,we prove that（?）_m（f）is a multiplicative function of m by using Chinese remainder theorem,i.e.,（?）_m（f）=Πp|m（?）pνp（m）（f）,where νp（m）stands for the padic valuation of m.Let f（x）=∑j=1r akjx（x-1）...（x-kj+1）be a polynomial of degree kr,where k1<k2<...<kr are positive integers,all akj are nonzero integers.Define the truncated polynomial of f by f_s（x）:=∑j=1 r-s akjx（x-1）...（x-kj+1）for 0 ≤s ≤r-1,f-1（x）:=0 and fr（x）:=1.By making use of p-adic analysis,we show that（?）pn（f）=pn-ε,where ε=min{n,min{νp（f’_s（a））:1≤a≤pn}},where s is uniquely determined by deg（f_s）<μp（n）:=min{t≥0:νp（t!）≥n}≤deg（f_s-1）and f’_s is the derivation of f_s.This solves Question（1）.On the other hand,with the help of Smith norm form over the ring Z/pnZ and the arithmetic function μp,we arrive at a formula for the number of polynomial functions f with a given smallest positive period.Thus Question（2）is solved.The third topic of this thesis is the Gawron-Miska-Ulas conjecture about the coefficients of the power series expansion of （?） In 2018,Gawron,Miska and Ulas together with Schinzel proved that the sequence{t2（n）}n=1∞ contains all nonzero integers.When m≥3,Gawron,Miska and Ulas conjectured that there are infinitely many positive integers do not appear in the sequence {t2（n）n=1∞.In Chapter 3,we use the congruence method and p-adic analysis to confirm the truth of this conjecture for the case of m being a power of an odd prime.