I want to know if Hilbert's Nullstellensatz can be used to compute the greatest common divisor/factor (GCD) with a known degree for a set of polynomials without common zeros.
The following is the formal setting:
Let $P_1, P_2, \dots, P_k$ be a set of polynomials in $\mathbb{C}[x_1, x_2, \dots, x_n]=\mathbb{C}[X]$ without common zeros. Let $G$ be a polynomial in $\mathbb{C}[X]$ with a known degree $d \geq 1$. If I am given $P_1G, P_2G, \dots, P_kG$, can I estimate $G$ up to a constant factor using Hilbert's Nullstellensatz?
Here is what I have so far:
Hilbert's Nullstellensatz says there are polynomials $R_1, R_2, \dots, R_k$ in $\mathbb{C}[X]$ that give $$\begin{align}P_1R_1 + P_2R_2 + \dots + P_kR_k = 1. \tag{1}\label{eq1}\end{align}$$ If $P_1, P_2, \dots, P_k$ are given, then $R_1, R_2, \dots, R_k$ can be found by solving a system of linear equations. More specifically, we can hypothetically set the maximum degree of $R_1, R_2, \dots, R_k$ to be $r$ and write out a system of linear equations with unknowns being the coefficients of $R_i$s. Then we can set 1) the equation that corresponds to the constant terms equal to $1$ and 2) all other equations that correspond to degree $\geq 1$ equal to $0$. If this system has a solution, then we have found a Nullstellensatz certificate $R_1, R_2, \dots, R_k$. If not, then we can increase the degree $r$ to a higher number and repeat the process. Bounds on the degree $d$ are known and has been studied in the form of effective Nullstellensatz.
Now multiplying $(1)$ with $G$ we have $$\begin{align}(P_1G)R_1 + (P_2G)R_2 + \dots + (P_kG)R_k = G \tag{2}\label{eq2}\end{align}.$$ This means that the exact same certificate $R_1, R_2, \dots, R_k$ that satisfy Eq. 1 would give us the greatest common factor $G$ from $P_1G, P_2G, \dots, P_kG$ in Eq. 2. (Question 1) Is this statement true? or am I being too naive?
I have attempted to find $R_1, R_2, \dots, R_k$ from Eq. 2 by the same method in Eq. 1. Namely, I set the equations that correspond to degree $> d$ equal $0$, which is a variant of 2) above. However, (Question 2) I am not sure how to setup the equivalent of 1) in this case.
Am I on the right track to tackle this problem? Any comments to present the questions in a better form are greatly appreciated.