- #1
mathmari
Gold Member
MHB
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Hey! 
One part of the paper that I am reading is the following:
Let $R$ be a commutative ring with unity and let $R'$ be a subring of $R$.
We say that the diophantine problem for $R$ with coefficients in $R'$ is unsolvable (solvable) if there exists no (an) algorithm to decide whether or not a polynomial equation (in several variables) with coefficients in $R'$ has a solution in $R$.
$$\dots \dots \dots \dots \dots$$
Theorem.
Let $R$ be an integral domain of characteristic zero; then the diophantine problem for $R[T]$ with coefficients in $\mathbb{Z}[T]$ is unsolvable.
($R[T]$ denotes the ring of polynomials over $R$, in one variable $T$.)
$$\dots \dots \dots \dots \dots$$
It is obvious that the diophantine problem for $R[T]$ with coefficients in $\mathbb{Z}$ is solvable if and only if the diophantine problem for $R$ with coefficients in $\mathbb{Z}$ is solvable.
Could you explain to me the last sentence?
Why does this stand?
Does the direction $\Leftarrow$ stand because of the following?
We know that there is an algorithm that decides whether or not an equation with coefficients in $\mathbb{Z}$ has a solution in $R$.
We consider this equation as the constant term of a polynomial equation, so there is an algorithm that decides whether or not an equation with coefficients in $\mathbb{Z}$ has a solution in $R[T]$.
Is the justification of this direction correct?
One part of the paper that I am reading is the following:
Let $R$ be a commutative ring with unity and let $R'$ be a subring of $R$.
We say that the diophantine problem for $R$ with coefficients in $R'$ is unsolvable (solvable) if there exists no (an) algorithm to decide whether or not a polynomial equation (in several variables) with coefficients in $R'$ has a solution in $R$.
$$\dots \dots \dots \dots \dots$$
Theorem.
Let $R$ be an integral domain of characteristic zero; then the diophantine problem for $R[T]$ with coefficients in $\mathbb{Z}[T]$ is unsolvable.
($R[T]$ denotes the ring of polynomials over $R$, in one variable $T$.)
$$\dots \dots \dots \dots \dots$$
It is obvious that the diophantine problem for $R[T]$ with coefficients in $\mathbb{Z}$ is solvable if and only if the diophantine problem for $R$ with coefficients in $\mathbb{Z}$ is solvable.
Could you explain to me the last sentence?
Why does this stand?
Does the direction $\Leftarrow$ stand because of the following?
We know that there is an algorithm that decides whether or not an equation with coefficients in $\mathbb{Z}$ has a solution in $R$.
We consider this equation as the constant term of a polynomial equation, so there is an algorithm that decides whether or not an equation with coefficients in $\mathbb{Z}$ has a solution in $R[T]$.
Is the justification of this direction correct?