Simply put, I have a few vectors of polynomials $\{d_i\}$ and I know one of their combinations gives $x$, which is also a vector of polynomials. How do I find the combinational coefficients $a_i$, namely, $$x=\sum_i a_id_i .$$
To be precise, the vectors of polynomials $\{d_i\}$ in question constitute the generating set of the kernel of a map $\psi$ (defined below), which is a (sub)module of the free module $M$ in polynomial ring $R$. Consider that they are known beforehand. Then for another element of the kernel, $x$, one needs to find the proper combinational coefficients.
I am not familiar with the ring theory either the module of syzygies. But I am doing some practical calculations which involve the first module of syzygies. For a given (multivariable) polynomial ring $R$, I understand that the first module of syzygies is related to the kernel of the following map (eg. consider the case with four generators) $$\psi: R^4\to R $$ $$c_1r_1+c_2r_2+c_3r_3+c_4r_4 \mapsto r$$ where $r_1,\cdots,r_4$ are four given elements of the polynomial ring, and the four coefficients $c_i$ are arbitrary coefficients, also polynomials. As discussed above, $r_i$ serve as the generators of an ideal, and by definition, the map constitutes a free module. Furthermore, the first module of syzygies is defined as the kernel of the above map, which is a submodule. My question is about how to find the expansion coefficients of an element $x$ belong to the kernel in terms of its generating set $\{d_i\}$. It seems to be a rather rudimental task but I have no idea how it might work. Many thanks in advance!
Below, I give explicitly what I need to solve and what I have tried. The generating set $\{d_j\}$ ($j=1,\cdots,6$) consists of six four-component ``vectors'' of polynomials in six variables $D_i$ and $D_{i'}$ ($i'$ is denoted by $Pi$ below) with $i=1,2,3$.
d1 = {(1 - Subscript[D, 1] Subscript[D, P1]), 0, 0, -(1 - Subscript[D, 1] Subscript[D, 2] Subscript[D, 3])};
d2 = {Subscript[D, 3] - Subscript[D, P1] Subscript[D, P2], -(Subscript[D, P1] - Subscript[D, 3] Subscript[D, 2]), 0, -Subscript[D, 3] (1 - Subscript[D, P2] Subscript[D, 2])};
d3 = {Subscript[D, 1] - Subscript[D, P3] Subscript[D, P2], -(Subscript[D, P3] - Subscript[D, 1] Subscript[D, 2]), -(1 - Subscript[D, P2] Subscript[D, 2]), 0};
d4 = {Subscript[D, 1] (Subscript[D, P3] Subscript[D, 3] - 1), 0, -(Subscript[D, 1] Subscript[D, 2] Subscript[D, 3] - 1), 0};
d5 = {0, -(Subscript[D, P1] Subscript[D, 1] - 1), 0, -(Subscript[D, 3] Subscript[D, 1] - Subscript[D, P2])};
d6 = {(1 - Subscript[D, P3] Subscript[D, 3]), 0, -(Subscript[D, P1] - Subscript[D, 2] Subscript[D, 3]), -(1 - Subscript[D, P3] Subscript[D, 3])};
Now I need to find the combinational coefficients $a_i$ so that $$x=\sum_{i=1}^6 a_i d_i ,$$ where
x = {Subscript[D, 2] Subscript[D, P3] Subscript[D, P3] - Subscript[D, 2] Subscript[D, P3] Subscript[D, P3] Subscript[D, P1] Subscript[D, 1] - Subscript[D, 2] Subscript[D, 1] Subscript[D, P1] Subscript[D, P3] Subscript[D, P3] + Subscript[D, 2] Subscript[D, 1] Subscript[D, P1] Subscript[D, P3] Subscript[D, P3] Subscript[D, P1] Subscript[D, 1], 0, -(Subscript[D, P3] Subscript[D, 2] Subscript[D, 2] - Subscript[D, P3] Subscript[D, 2] Subscript[D, 2] Subscript[D, 1] Subscript[D, P1] - Subscript[D, P3] Subscript[D, P1] Subscript[D, 1] Subscript[D, 2] Subscript[D, 2] + Subscript[D, P3] Subscript[D, P1] Subscript[D, 1] Subscript[D, 2] Subscript[D, 2] Subscript[D, 1] Subscript[D, P1]), -(Subscript[D, 2] Subscript[D, P3] Subscript[D, P3] - Subscript[D, 2] Subscript[D, P1] Subscript[D, 1] Subscript[D, P3] Subscript[D, P3] - Subscript[D, P3] Subscript[D, 2] Subscript[D, 2] Subscript[D, 1] + Subscript[D, P3] Subscript[D, P1] Subscript[D, 1]Subscript[D, 2] Subscript[D, 2] Subscript[D, 1])};
One may solve the above problem straightforwardly. But as explained above, the problem is closely associated with the one regarding the first module of syzygies, and I am not sure a general solution is related. Therefore I also present them here. Consider an ideal generated by following four polynomials
r1 = (-1 + Subscript[D, 1] Subscript[D, 2] Subscript[D, 3]);
r2 = (-1 + Subscript[D, 1] Subscript[D, P1]);
r3 = (-Subscript[D, 1] Subscript[D, 3] + Subscript[D, P2]);
r4 = (-Subscript[D, 1] + Subscript[D, 1] Subscript[D, 3] Subscript[D, P3]);
When linearly combined with four arbitrary polynomial coefficients $c_i$, the resultant expression $$\sum c_ir_i$$ is also a polynomial, which corresponds to a map $$R^4\to R .$$ Now, the kernel of the above map can be derived through the Grobner basis $G=\{g_i\}$ of these polynomials, which can be obtained (without awareness of the details) by the Mathematica command GroebnerBasis[].
GroebnerBasis[{(-1 + Subscript[D, 1] Subscript[D, 2] Subscript[D, 3]), (-1 + Subscript[D, 1] Subscript[D, P1]), (-Subscript[D, 1] Subscript[D, 3] + Subscript[D, P2]), (-Subscript[D, 1] + Subscript[D, 1] Subscript[D, 3] Subscript[D, P3])}, {Subscript[D, 1], Subscript[D, 2], Subscript[D, 3], Subscript[D, P1], Subscript[D, P2], Subscript[D, P3]}]
To obtain the coefficients $a_i$, I tried the command PolynomialReduce[] to expand the $x$ in both $D_i$ ($D_{i'}$) and individual vector component (assigned by artificial variable $A_i$), as follows
DSixBases = {d1[[1]] A1 + d1[[2]] A2 + d1[[3]] A3 + d1[[4]] A4, d2[[1]] A1 + d2[[2]] A2 + d2[[3]] A3 + d2[[4]] A4, d3[[1]] A1 + d3[[2]] A2 + d3[[3]] A3 + d3[[4]] A4, d4[[1]] A1 + d4[[2]] A2 + d4[[3]] A3 + d4[[4]] A4, d5[[1]] A1 + d5[[2]] A2 + d5[[3]] A3 + d5[[4]] A4, d6[[1]] A1 + d6[[2]] A2 + d6[[3]] A3 + d6[[4]] A4}
PolynomialReduce[x[[1]] A1 + x[[2]] A2 + x[[3]] A3 + x[[4]] A4, DSixBases, {Subscript[D, 1], Subscript[D, 2], Subscript[D, 3], Subscript[D, P1], Subscript[D, P2], Subscript[D, P3], A1, A2, A3, A4}]
However, it does not seem to work.
