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Suppose A is Diophantine Equations which have a variable x,such that every other variable has function of x the variable as it's solution.For example,a Diophantine equation in three variables has two variables y,z, $$ y = x^2+1$$ $$ z = 4x^6+2$$ as their solutions.

All Diophantine Equations like A form B, a special class of Diophantine Equations.

EDIT:suppose $$P_1(x,y_1,\cdots,y_n)=P_2(x,y_1,\cdots,y_n)=\cdots=P_k(x,y_1,\cdots,y_n)=0$$ is Diopantine equations ,and there are functions$$y_1=f_1(x),\cdots,y_n=f_n(x)$$,which satisify the Diopantine equations.

Question: is any function of x for solution of other variables of such Diophantine Equations algebraic? or are $$y_1=f_1(x),\cdots,y_n=f_n(x)$$ all algebraic?

I think I have gotten it:the functions of x for solution of other variables of such Diophantine Equations should be algebraic,and they(and x) are all able to be parametrized by automorphic function .Somehow I had been confused with another problem in other direction when I posted the question.A lot of thanks to every one who has comment on it.

Suppose A is Diophantine Equations which have a variable x,such that every other variable has function of x the variable as it's solution.For example,a Diophantine equation in three variables has two variables y,z, $$ y = x^2+1$$ $$ z = 4x^6+2$$ as their solutions.

All Diophantine Equations like A form B, a special class of Diophantine Equations.

EDIT:suppose $$P_1(x,y_1,\cdots,y_n)=P_2(x,y_1,\cdots,y_n)=\cdots=P_k(x,y_1,\cdots,y_n)=0$$ is Diopantine equations ,and there are functions$$y_1=f_1(x),\cdots,y_n=f_n(x)$$,which satisify the Diopantine equations.

Question: is any function of x for solution of other variables of such Diophantine Equations algebraic? or are $$y_1=f_1(x),\cdots,y_n=f_n(x)$$ all algebraic?

Suppose A is Diophantine Equations which have a variable x,such that every other variable has function of x the variable as it's solution.For example,a Diophantine equation in three variables has two variables y,z, $$ y = x^2+1$$ $$ z = 4x^6+2$$ as their solutions.

All Diophantine Equations like A form B, a special class of Diophantine Equations.

EDIT:suppose $$P_1(x,y_1,\cdots,y_n)=P_2(x,y_1,\cdots,y_n)=\cdots=P_k(x,y_1,\cdots,y_n)=0$$ is Diopantine equations ,and there are functions$$y_1=f_1(x),\cdots,y_n=f_n(x)$$,which satisify the Diopantine equations.

Question: is any function of x for solution of other variables of such Diophantine Equations algebraic? or are $$y_1=f_1(x),\cdots,y_n=f_n(x)$$ all algebraic?

I think I have gotten it:the functions of x for solution of other variables of such Diophantine Equations should be algebraic,and they(and x) are all able to be parametrized by automorphic function .Somehow I had been confused with another problem in other direction when I posted the question

Suppose A is Diophantine Equations which have a variable x,such that every other variable has function of x the variable as it's solution.For example,a Diophantine equation in three variables has two variables y,z, $$ y = x^2+1$$ $$ z = 4x^6+2$$ as their solutions.

All Diophantine Equations like A form B, a special class of Diophantine Equations.

EDIT:suppose $$P_1(x,y_1,\cdots,y_n)=P_2(x,y_1,\cdots,y_n)=\cdots=P_k(x,y_1,\cdots,y_n)=0$$ is Diopantine equations ,and there are functions$$y_1=f_1(x),\cdots,y_n=f_n(x)$$,which satisify the Diopantine equations.

Question: is any function of x for solution of other variables of such Diophantine Equations algebraic? or are $$y_1=f_1(x),\cdots,y_n=f_n(x)$$ all algebraic?

I think I have gotten it:the functions of x for solution of other variables of such Diophantine Equations should be algebraic,and they(and x) are all able to be parametrized by automorphic function .Somehow I had been confused with another problem in other direction when I posted the question.A lot of thanks to every one who has comment on it.

Suppose A is Diophantine Equations which have a variable x,such that every other variable has function of x the variable as it's solution.For example,a Diophantine equation in three variables has two variables y,z, $$ y = x^2+1$$ $$ z = 4x^6+2$$ as their solutions.

All Diophantine Equations like A form B, a special class of Diophantine Equations.

EDIT:suppose $$P_1(x,y_1,\cdots,y_n)=P_2(x,y_1,\cdots,y_n)=\cdots=P_k(x,y_1,\cdots,y_n)=0$$ is Diopantine equations ,and there are functions$$y_1=f_1(x),\cdots,y_k=f_k(x)$$,which satisify the Diopantine equations.

Question: is any function of x for solution of other variables of such Diophantine Equations algebraic? or are $$y_1=f_1(x),\cdots,y_k=f_k(x)$$ all algebraic?

I think I have gotten it:the functions of x for solution of other variables of such Diophantine Equations should be algebraic,and they are all able to be parametrized by automorphic function.Somehow I had been confused with another problem in other direction when I posted the question

Suppose A is Diophantine Equations which have a variable x,such that every other variable has function of x the variable as it's solution.For example,a Diophantine equation in three variables has two variables y,z, $$ y = x^2+1$$ $$ z = 4x^6+2$$ as their solutions.

All Diophantine Equations like A form B, a special class of Diophantine Equations.

EDIT:suppose $$P_1(x,y_1,\cdots,y_n)=P_2(x,y_1,\cdots,y_n)=\cdots=P_k(x,y_1,\cdots,y_n)=0$$ is Diopantine equations ,and there are functions$$y_1=f_1(x),\cdots,y_n=f_n(x)$$,which satisify the Diopantine equations.

Question: is any function of x for solution of other variables of such Diophantine Equations algebraic? or are $$y_1=f_1(x),\cdots,y_n=f_n(x)$$ all algebraic?

I think I have gotten it:the functions of x for solution of other variables of such Diophantine Equations should be algebraic,and they(and x) are all able to be parametrized by automorphic function  .Somehow I had been confused with another problem in other direction when I posted the question