Computing the resultant of two polynomials of degree $4$

Question

Let $f=f(t)=\sum_{i=0}^{4}a_ix^i ,g=g(t)=\sum_{j=0}^{4}b_jx^j \in L[x]$, $L$ is a commutative ring. I would like to compute the resultant of $f$ and $g$.

I wish to check a few examples in low dimensions, but even in low dimension (with $\deg(f)=\deg(g)$) such as four, the computation is quite lengthy-- the matrix is of order $8$.

I know that there are computer algebra systems that can do these calculations, but unfortunately I have not succeeded to install Macaulay2 on my computer, so I ask for help here; see the answer to this question.

What is the resultant of $f$ and $g$ as above?

I really apologize if my question is not so appropriate (I would prefer to get help in installing Macaulay2 or a link where one can compute the resultant on-line, but asking for that seems totally not appropriate to this site, and I do not know where to ask these questions).

Thank you very much!

Just curious, what could the exact expression be useful for? You can calculate it online in WA for example resultant[ a_0 + a_1 x + a_2 x^2 + a_3 x^3 + a_4 x^4, b_0 + b_1 x + b_2 x^2 + b_3 x^3 + b_4 x^4 , x ] but it's certainly not pretty (in the general case). — dxiv, Jun 28 '18 at 03:37
I wish to find a condition on the coefficients of $f$ and $g$ that will imply that the resultant is a nonzero 'scalar'. More precisely, if $L=k[y]$ (so the resultant belongs to $k[y]$), $k$ is a field, I wish that the resultant will belong to $k$. — user237522, Jun 28 '18 at 03:57
Just a thought, but if that's a step towards proving/finding something else, maybe there is another way to get there, more lightweight than resultants. — dxiv, Jun 28 '18 at 04:07
Thank you for your advice (and thanks again for the link; it's very helpful!). — user237522, Jun 28 '18 at 04:11

score 2 · Accepted Answer · answered Jun 28 '18 at 03:42

Using Maple, there are $219$ terms: $$ \eqalign{&{a_{{0}}}^{4}{b_{{4}}}^{4}-{a_{{0}}}^{3}a_{{1}}b_{{3}}{b_{{4}}}^{3}-2 \,{a_{{0}}}^{3}a_{{2}}b_{{2}}{b_{{4}}}^{3}+{a_{{0}}}^{3}a_{{2}}{b_{{3} }}^{2}{b_{{4}}}^{2}-3\,{a_{{0}}}^{3}a_{{3}}b_{{1}}{b_{{4}}}^{3}+3\,{a_ {{0}}}^{3}a_{{3}}b_{{2}}b_{{3}}{b_{{4}}}^{2}-{a_{{0}}}^{3}a_{{3}}{b_{{ 3}}}^{3}b_{{4}} \cr & -4\,{a_{{0}}}^{3}a_{{4}}b_{{0}}{b_{{4}}}^{3}+4\,{a_{{0}}}^{3}a_{{4}}b_ {{1}}b_{{3}}{b_{{4}}}^{2}+2\,{a_{{0}}}^{3}a_{{4}}{b_{{2}}}^{2}{b_{{4}} }^{2}-4\,{a_{{0}}}^{3}a_{{4}}b_{{2}}{b_{{3}}}^{2}b_{{4}}+{a_{{0}}}^{3} a_{{4}}{b_{{3}}}^{4}+{a_{{0}}}^{2}{a_{{1}}}^{2}b_{{2}}{b_{{4}}}^{3}+3 \,{a_{{0}}}^{2}a_{{1}}a_{{2}}b_{{1}}{b_{{4}}}^{3} \cr & -{a_{{0}}}^{2}a_{{1}}a_{{2}}b_{{2}}b_{{3}}{b_{{4}}}^{2}+4\,{a_{{0}}}^{ 2}a_{{1}}a_{{3}}b_{{0}}{b_{{4}}}^{3}-{a_{{0}}}^{2}a_{{1}}a_{{3}}b_{{1} }b_{{3}}{b_{{4}}}^{2}-2\,{a_{{0}}}^{2}a_{{1}}a_{{3}}{b_{{2}}}^{2}{b_{{ 4}}}^{2}+{a_{{0}}}^{2}a_{{1}}a_{{3}}b_{{2}}{b_{{3}}}^{2}b_{{4}}-{a_{{0 }}}^{2}a_{{1}}a_{{4}}b_{{0}}b_{{3}}{b_{{4}}}^{2}-5\,{a_{{0}}}^{2}a_{{1 }}a_{{4}}b_{{1}}b_{{2}}{b_{{4}}}^{2} \cr & +{a_{{0}}}^{2}a_{{1}}a_{{4}}b_{{1}}{b_{{3}}}^{2}b_{{4}}+3\,{a_{{0}}}^{2 }a_{{1}}a_{{4}}{b_{{2}}}^{2}b_{{3}}b_{{4}}-{a_{{0}}}^{2}a_{{1}}a_{{4}} b_{{2}}{b_{{3}}}^{3}+2\,{a_{{0}}}^{2}{a_{{2}}}^{2}b_{{0}}{b_{{4}}}^{3} -2\,{a_{{0}}}^{2}{a_{{2}}}^{2}b_{{1}}b_{{3}}{b_{{4}}}^{2}+{a_{{0}}}^{2 }{a_{{2}}}^{2}{b_{{2}}}^{2}{b_{{4}}}^{2}-5\,{a_{{0}}}^{2}a_{{2}}a_{{3} }b_{{0}}b_{{3}}{b_{{4}}}^{2} \cr & +{a_{{0}}}^{2}a_{{2}}a_{{3}}b_{{1}}b_{{2}}{b_{{4}}}^{2}+2\,{a_{{0}}}^{2 }a_{{2}}a_{{3}}b_{{1}}{b_{{3}}}^{2}b_{{4}}-{a_{{0}}}^{2}a_{{2}}a_{{3}} {b_{{2}}}^{2}b_{{3}}b_{{4}}+2\,{a_{{0}}}^{2}a_{{2}}a_{{4}}b_{{0}}b_{{2 }}{b_{{4}}}^{2}+2\,{a_{{0}}}^{2}a_{{2}}a_{{4}}b_{{0}}{b_{{3}}}^{2}b_{{ 4}}-3\,{a_{{0}}}^{2}a_{{2}}a_{{4}}{b_{{1}}}^{2}{b_{{4}}}^{2}+4\,{a_{{0 }}}^{2}a_{{2}}a_{{4}}b_{{1}}b_{{2}}b_{{3}}b_{{4}} \cr & -2\,{a_{{0}}}^{2}a_{{2}}a_{{4}}b_{{1}}{b_{{3}}}^{3}-2\,{a_{{0}}}^{2}a_ {{2}}a_{{4}}{b_{{2}}}^{3}b_{{4}}+{a_{{0}}}^{2}a_{{2}}a_{{4}}{b_{{2}}}^ {2}{b_{{3}}}^{2}-3\,{a_{{0}}}^{2}{a_{{3}}}^{2}b_{{0}}b_{{2}}{b_{{4}}}^ {2}+3\,{a_{{0}}}^{2}{a_{{3}}}^{2}b_{{0}}{b_{{3}}}^{2}b_{{4}}+3\,{a_{{0 }}}^{2}{a_{{3}}}^{2}{b_{{1}}}^{2}{b_{{4}}}^{2}-3\,{a_{{0}}}^{2}{a_{{3} }}^{2}b_{{1}}b_{{2}}b_{{3}}b_{{4}} \cr & +{a_{{0}}}^{2}{a_{{3}}}^{2}{b_{{2}}}^{3}b_{{4}}+5\,{a_{{0}}}^{2}a_{{3}} a_{{4}}b_{{0}}b_{{1}}{b_{{4}}}^{2}+2\,{a_{{0}}}^{2}a_{{3}}a_{{4}}b_{{0 }}b_{{2}}b_{{3}}b_{{4}}-3\,{a_{{0}}}^{2}a_{{3}}a_{{4}}b_{{0}}{b_{{3}}} ^{3}-5\,{a_{{0}}}^{2}a_{{3}}a_{{4}}{b_{{1}}}^{2}b_{{3}}b_{{4}}+{a_{{0} }}^{2}a_{{3}}a_{{4}}b_{{1}}{b_{{2}}}^{2}b_{{4}}+3\,{a_{{0}}}^{2}a_{{3} }a_{{4}}b_{{1}}b_{{2}}{b_{{3}}}^{2} \cr & -{a_{{0}}}^{2}a_{{3}}a_{{4}}{b_{{2}}}^{3}b_{{3}}+6\,{a_{{0}}}^{2}{a_{{ 4}}}^{2}{b_{{0}}}^{2}{b_{{4}}}^{2}-8\,{a_{{0}}}^{2}{a_{{4}}}^{2}b_{{0} }b_{{1}}b_{{3}}b_{{4}}-4\,{a_{{0}}}^{2}{a_{{4}}}^{2}b_{{0}}{b_{{2}}}^{ 2}b_{{4}}+4\,{a_{{0}}}^{2}{a_{{4}}}^{2}b_{{0}}b_{{2}}{b_{{3}}}^{2}+4\, {a_{{0}}}^{2}{a_{{4}}}^{2}{b_{{1}}}^{2}b_{{2}}b_{{4}}+2\,{a_{{0}}}^{2} {a_{{4}}}^{2}{b_{{1}}}^{2}{b_{{3}}}^{2} \cr & -4\,{a_{{0}}}^{2}{a_{{4}}}^{2}b_{{1}}{b_{{2}}}^{2}b_{{3}}+{a_{{0}}}^{2 }{a_{{4}}}^{2}{b_{{2}}}^{4}-a_{{0}}{a_{{1}}}^{3}b_{{1}}{b_{{4}}}^{3}-4 \,a_{{0}}{a_{{1}}}^{2}a_{{2}}b_{{0}}{b_{{4}}}^{3}+a_{{0}}{a_{{1}}}^{2} a_{{2}}b_{{1}}b_{{3}}{b_{{4}}}^{2}+a_{{0}}{a_{{1}}}^{2}a_{{3}}b_{{0}}b _{{3}}{b_{{4}}}^{2}+2\,a_{{0}}{a_{{1}}}^{2}a_{{3}}b_{{1}}b_{{2}}{b_{{4 }}}^{2} \cr & -a_{{0}}{a_{{1}}}^{2}a_{{3}}b_{{1}}{b_{{3}}}^{2}b_{{4}}+2\,a_{{0}}{a_{ {1}}}^{2}a_{{4}}b_{{0}}b_{{2}}{b_{{4}}}^{2}-a_{{0}}{a_{{1}}}^{2}a_{{4} }b_{{0}}{b_{{3}}}^{2}b_{{4}}+3\,a_{{0}}{a_{{1}}}^{2}a_{{4}}{b_{{1}}}^{ 2}{b_{{4}}}^{2}-3\,a_{{0}}{a_{{1}}}^{2}a_{{4}}b_{{1}}b_{{2}}b_{{3}}b_{ {4}}+a_{{0}}{a_{{1}}}^{2}a_{{4}}b_{{1}}{b_{{3}}}^{3}+3\,a_{{0}}a_{{1}} {a_{{2}}}^{2}b_{{0}}b_{{3}}{b_{{4}}}^{2} \cr & -a_{{0}}a_{{1}}{a_{{2}}}^{2}b_{{1}}b_{{2}}{b_{{4}}}^{2}+4\,a_{{0}}a_{{ 1}}a_{{2}}a_{{3}}b_{{0}}b_{{2}}{b_{{4}}}^{2}-3\,a_{{0}}a_{{1}}a_{{2}}a _{{3}}b_{{0}}{b_{{3}}}^{2}b_{{4}}-3\,a_{{0}}a_{{1}}a_{{2}}a_{{3}}{b_{{ 1}}}^{2}{b_{{4}}}^{2}+a_{{0}}a_{{1}}a_{{2}}a_{{3}}b_{{1}}b_{{2}}b_{{3} }b_{{4}}+2\,a_{{0}}a_{{1}}a_{{2}}a_{{4}}b_{{0}}b_{{1}}{b_{{4}}}^{2}-8 \,a_{{0}}a_{{1}}a_{{2}}a_{{4}}b_{{0}}b_{{2}}b_{{3}}b_{{4}} \cr & +3\,a_{{0}}a_{{1}}a_{{2}}a_{{4}}b_{{0}}{b_{{3}}}^{3}+a_{{0}}a_{{1}}a_{{ 2}}a_{{4}}{b_{{1}}}^{2}b_{{3}}b_{{4}}+2\,a_{{0}}a_{{1}}a_{{2}}a_{{4}}b _{{1}}{b_{{2}}}^{2}b_{{4}}-a_{{0}}a_{{1}}a_{{2}}a_{{4}}b_{{1}}b_{{2}}{ b_{{3}}}^{2}-5\,a_{{0}}a_{{1}}{a_{{3}}}^{2}b_{{0}}b_{{1}}{b_{{4}}}^{2} +a_{{0}}a_{{1}}{a_{{3}}}^{2}b_{{0}}b_{{2}}b_{{3}}b_{{4}}+2\,a_{{0}}a_{ {1}}{a_{{3}}}^{2}{b_{{1}}}^{2}b_{{3}}b_{{4}} \cr & -a_{{0}}a_{{1}}{a_{{3}}}^{2}b_{{1}}{b_{{2}}}^{2}b_{{4}}-8\,a_{{0}}a_{{ 1}}a_{{3}}a_{{4}}{b_{{0}}}^{2}{b_{{4}}}^{2}+10\,a_{{0}}a_{{1}}a_{{3}}a _{{4}}b_{{0}}b_{{1}}b_{{3}}b_{{4}}-a_{{0}}a_{{1}}a_{{3}}a_{{4}}b_{{0}} b_{{2}}{b_{{3}}}^{2}-a_{{0}}a_{{1}}a_{{3}}a_{{4}}{b_{{1}}}^{2}b_{{2}}b _{{4}}-2\,a_{{0}}a_{{1}}a_{{3}}a_{{4}}{b_{{1}}}^{2}{b_{{3}}}^{2}+a_{{0 }}a_{{1}}a_{{3}}a_{{4}}b_{{1}}{b_{{2}}}^{2}b_{{3}} \cr & +5\,a_{{0}}a_{{1}}{a_{{4}}}^{2}{b_{{0}}}^{2}b_{{3}}b_{{4}}+2\,a_{{0}}a_ {{1}}{a_{{4}}}^{2}b_{{0}}b_{{1}}b_{{2}}b_{{4}}-5\,a_{{0}}a_{{1}}{a_{{4 }}}^{2}b_{{0}}b_{{1}}{b_{{3}}}^{2}+a_{{0}}a_{{1}}{a_{{4}}}^{2}b_{{0}}{ b_{{2}}}^{2}b_{{3}}-3\,a_{{0}}a_{{1}}{a_{{4}}}^{2}{b_{{1}}}^{3}b_{{4}} +3\,a_{{0}}a_{{1}}{a_{{4}}}^{2}{b_{{1}}}^{2}b_{{2}}b_{{3}}-a_{{0}}a_{{ 1}}{a_{{4}}}^{2}b_{{1}}{b_{{2}}}^{3} \cr & -2\,a_{{0}}{a_{{2}}}^{3}b_{{0}}b_{{2}}{b_{{4}}}^{2}+a_{{0}}{a_{{2}}}^{ 3}{b_{{1}}}^{2}{b_{{4}}}^{2}+a_{{0}}{a_{{2}}}^{2}a_{{3}}b_{{0}}b_{{1}} {b_{{4}}}^{2}+2\,a_{{0}}{a_{{2}}}^{2}a_{{3}}b_{{0}}b_{{2}}b_{{3}}b_{{4 }}-a_{{0}}{a_{{2}}}^{2}a_{{3}}{b_{{1}}}^{2}b_{{3}}b_{{4}}-4\,a_{{0}}{a _{{2}}}^{2}a_{{4}}{b_{{0}}}^{2}{b_{{4}}}^{2}+4\,a_{{0}}{a_{{2}}}^{2}a_ {{4}}b_{{0}}{b_{{2}}}^{2}b_{{4}} \cr & -2\,a_{{0}}{a_{{2}}}^{2}a_{{4}}b_{{0}}b_{{2}}{b_{{3}}}^{2}-2\,a_{{0}}{ a_{{2}}}^{2}a_{{4}}{b_{{1}}}^{2}b_{{2}}b_{{4}}+a_{{0}}{a_{{2}}}^{2}a_{ {4}}{b_{{1}}}^{2}{b_{{3}}}^{2}+4\,a_{{0}}a_{{2}}{a_{{3}}}^{2}{b_{{0}}} ^{2}{b_{{4}}}^{2}-a_{{0}}a_{{2}}{a_{{3}}}^{2}b_{{0}}b_{{1}}b_{{3}}b_{{ 4}}-2\,a_{{0}}a_{{2}}{a_{{3}}}^{2}b_{{0}}{b_{{2}}}^{2}b_{{4}}+a_{{0}}a _{{2}}{a_{{3}}}^{2}{b_{{1}}}^{2}b_{{2}}b_{{4}} \cr & +2\,a_{{0}}a_{{2}}a_{{3}}a_{{4}}{b_{{0}}}^{2}b_{{3}}b_{{4}}-8\,a_{{0}}a _{{2}}a_{{3}}a_{{4}}b_{{0}}b_{{1}}b_{{2}}b_{{4}}+a_{{0}}a_{{2}}a_{{3}} a_{{4}}b_{{0}}b_{{1}}{b_{{3}}}^{2}+2\,a_{{0}}a_{{2}}a_{{3}}a_{{4}}b_{{0 }}{b_{{2}}}^{2}b_{{3}}+3\,a_{{0}}a_{{2}}a_{{3}}a_{{4}}{b_{{1}}}^{3}b_{ {4}}-a_{{0}}a_{{2}}a_{{3}}a_{{4}}{b_{{1}}}^{2}b_{{2}}b_{{3}}+2\,a_{{0} }a_{{2}}{a_{{4}}}^{2}{b_{{0}}}^{2}b_{{2}}b_{{4}} \cr & -3\,a_{{0}}a_{{2}}{a_{{4}}}^{2}{b_{{0}}}^{2}{b_{{3}}}^{2}+2\,a_{{0}}a_ {{2}}{a_{{4}}}^{2}b_{{0}}{b_{{1}}}^{2}b_{{4}}+4\,a_{{0}}a_{{2}}{a_{{4} }}^{2}b_{{0}}b_{{1}}b_{{2}}b_{{3}}-2\,a_{{0}}a_{{2}}{a_{{4}}}^{2}b_{{0 }}{b_{{2}}}^{3}-2\,a_{{0}}a_{{2}}{a_{{4}}}^{2}{b_{{1}}}^{3}b_{{3}}+a_{ {0}}a_{{2}}{a_{{4}}}^{2}{b_{{1}}}^{2}{b_{{2}}}^{2}-3\,a_{{0}}{a_{{3}}} ^{3}{b_{{0}}}^{2}b_{{3}}b_{{4}} \cr & +3\,a_{{0}}{a_{{3}}}^{3}b_{{0}}b_{{1}}b_{{2}}b_{{4}}-a_{{0}}{a_{{3}}}^{ 3}{b_{{1}}}^{3}b_{{4}}+2\,a_{{0}}{a_{{3}}}^{2}a_{{4}}{b_{{0}}}^{2}b_{{ 2}}b_{{4}}+3\,a_{{0}}{a_{{3}}}^{2}a_{{4}}{b_{{0}}}^{2}{b_{{3}}}^{2}-a_ {{0}}{a_{{3}}}^{2}a_{{4}}b_{{0}}{b_{{1}}}^{2}b_{{4}}-3\,a_{{0}}{a_{{3} }}^{2}a_{{4}}b_{{0}}b_{{1}}b_{{2}}b_{{3}}+a_{{0}}{a_{{3}}}^{2}a_{{4}}{ b_{{1}}}^{3}b_{{3}} \cr & -a_{{0}}a_{{3}}{a_{{4}}}^{2}{b_{{0}}}^{2}b_{{1}}b_{{4}}-5\,a_{{0}}a_{{ 3}}{a_{{4}}}^{2}{b_{{0}}}^{2}b_{{2}}b_{{3}}+a_{{0}}a_{{3}}{a_{{4}}}^{2 }b_{{0}}{b_{{1}}}^{2}b_{{3}}+3\,a_{{0}}a_{{3}}{a_{{4}}}^{2}b_{{0}}b_{{ 1}}{b_{{2}}}^{2}-a_{{0}}a_{{3}}{a_{{4}}}^{2}{b_{{1}}}^{3}b_{{2}}-4\,a_ {{0}}{a_{{4}}}^{3}{b_{{0}}}^{3}b_{{4}}+4\,a_{{0}}{a_{{4}}}^{3}{b_{{0}} }^{2}b_{{1}}b_{{3}} \cr & +2\,a_{{0}}{a_{{4}}}^{3}{b_{{0}}}^{2}{b_{{2}}}^{2}-4\,a_{{0}}{a_{{4}}}^ {3}b_{{0}}{b_{{1}}}^{2}b_{{2}}+a_{{0}}{a_{{4}}}^{3}{b_{{1}}}^{4}+{a_{{ 1}}}^{4}b_{{0}}{b_{{4}}}^{3}-{a_{{1}}}^{3}a_{{2}}b_{{0}}b_{{3}}{b_{{4} }}^{2}-2\,{a_{{1}}}^{3}a_{{3}}b_{{0}}b_{{2}}{b_{{4}}}^{2}+{a_{{1}}}^{3 }a_{{3}}b_{{0}}{b_{{3}}}^{2}b_{{4}} \cr & -3\,{a_{{1}}}^{3}a_{{4}}b_{{0}}b_{{1}}{b_{{4}}}^{2}+3\,{a_{{1}}}^{3}a_ {{4}}b_{{0}}b_{{2}}b_{{3}}b_{{4}}-{a_{{1}}}^{3}a_{{4}}b_{{0}}{b_{{3}}} ^{3}+{a_{{1}}}^{2}{a_{{2}}}^{2}b_{{0}}b_{{2}}{b_{{4}}}^{2}+3\,{a_{{1}} }^{2}a_{{2}}a_{{3}}b_{{0}}b_{{1}}{b_{{4}}}^{2}-{a_{{1}}}^{2}a_{{2}}a_{ {3}}b_{{0}}b_{{2}}b_{{3}}b_{{4}}+4\,{a_{{1}}}^{2}a_{{2}}a_{{4}}{b_{{0} }}^{2}{b_{{4}}}^{2} \cr & -{a_{{1}}}^{2}a_{{2}}a_{{4}}b_{{0}}b_{{1}}b_{{3}}b_{{4}}-2\,{a_{{1}}}^ {2}a_{{2}}a_{{4}}b_{{0}}{b_{{2}}}^{2}b_{{4}}+{a_{{1}}}^{2}a_{{2}}a_{{4 }}b_{{0}}b_{{2}}{b_{{3}}}^{2}+2\,{a_{{1}}}^{2}{a_{{3}}}^{2}{b_{{0}}}^{ 2}{b_{{4}}}^{2}-2\,{a_{{1}}}^{2}{a_{{3}}}^{2}b_{{0}}b_{{1}}b_{{3}}b_{{ 4}}+{a_{{1}}}^{2}{a_{{3}}}^{2}b_{{0}}{b_{{2}}}^{2}b_{{4}}-5\,{a_{{1}}} ^{2}a_{{3}}a_{{4}}{b_{{0}}}^{2}b_{{3}}b_{{4}} \cr & +{a_{{1}}}^{2}a_{{3}}a_{{4}}b_{{0}}b_{{1}}b_{{2}}b_{{4}}+2\,{a_{{1}}}^{ 2}a_{{3}}a_{{4}}b_{{0}}b_{{1}}{b_{{3}}}^{2}-{a_{{1}}}^{2}a_{{3}}a_{{4} }b_{{0}}{b_{{2}}}^{2}b_{{3}}-3\,{a_{{1}}}^{2}{a_{{4}}}^{2}{b_{{0}}}^{2 }b_{{2}}b_{{4}}+3\,{a_{{1}}}^{2}{a_{{4}}}^{2}{b_{{0}}}^{2}{b_{{3}}}^{2 }+3\,{a_{{1}}}^{2}{a_{{4}}}^{2}b_{{0}}{b_{{1}}}^{2}b_{{4}}-3\,{a_{{1}} }^{2}{a_{{4}}}^{2}b_{{0}}b_{{1}}b_{{2}}b_{{3}} \cr & +{a_{{1}}}^{2}{a_{{4}}}^{2}b_{{0}}{b_{{2}}}^{3}-a_{{1}}{a_{{2}}}^{3}b_{ {0}}b_{{1}}{b_{{4}}}^{2}-4\,a_{{1}}{a_{{2}}}^{2}a_{{3}}{b_{{0}}}^{2}{b _{{4}}}^{2}+a_{{1}}{a_{{2}}}^{2}a_{{3}}b_{{0}}b_{{1}}b_{{3}}b_{{4}}+a_ {{1}}{a_{{2}}}^{2}a_{{4}}{b_{{0}}}^{2}b_{{3}}b_{{4}}+2\,a_{{1}}{a_{{2} }}^{2}a_{{4}}b_{{0}}b_{{1}}b_{{2}}b_{{4}}-a_{{1}}{a_{{2}}}^{2}a_{{4}}b _{{0}}b_{{1}}{b_{{3}}}^{2} \cr & +3\,a_{{1}}a_{{2}}{a_{{3}}}^{2}{b_{{0}}}^{2}b_{{3}}b_{{4}}-a_{{1}}a_{{2 }}{a_{{3}}}^{2}b_{{0}}b_{{1}}b_{{2}}b_{{4}}+4\,a_{{1}}a_{{2}}a_{{3}}a_ {{4}}{b_{{0}}}^{2}b_{{2}}b_{{4}}-3\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}{b_{{0 }}}^{2}{b_{{3}}}^{2}-3\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}b_{{0}}{b_{{1}}}^{ 2}b_{{4}}+a_{{1}}a_{{2}}a_{{3}}a_{{4}}b_{{0}}b_{{1}}b_{{2}}b_{{3}}-5\, a_{{1}}a_{{2}}{a_{{4}}}^{2}{b_{{0}}}^{2}b_{{1}}b_{{4}} \cr & +a_{{1}}a_{{2}}{a_{{4}}}^{2}{b_{{0}}}^{2}b_{{2}}b_{{3}}+2\,a_{{1}}a_{{2 }}{a_{{4}}}^{2}b_{{0}}{b_{{1}}}^{2}b_{{3}}-a_{{1}}a_{{2}}{a_{{4}}}^{2} b_{{0}}b_{{1}}{b_{{2}}}^{2}-2\,a_{{1}}{a_{{3}}}^{3}{b_{{0}}}^{2}b_{{2} }b_{{4}}+a_{{1}}{a_{{3}}}^{3}b_{{0}}{b_{{1}}}^{2}b_{{4}}+a_{{1}}{a_{{3 }}}^{2}a_{{4}}{b_{{0}}}^{2}b_{{1}}b_{{4}}+2\,a_{{1}}{a_{{3}}}^{2}a_{{4 }}{b_{{0}}}^{2}b_{{2}}b_{{3}} \cr & -a_{{1}}{a_{{3}}}^{2}a_{{4}}b_{{0}}{b_{{1}}}^{2}b_{{3}}+4\,a_{{1}}a_{{ 3}}{a_{{4}}}^{2}{b_{{0}}}^{3}b_{{4}}-a_{{1}}a_{{3}}{a_{{4}}}^{2}{b_{{0 }}}^{2}b_{{1}}b_{{3}}-2\,a_{{1}}a_{{3}}{a_{{4}}}^{2}{b_{{0}}}^{2}{b_{{ 2}}}^{2}+a_{{1}}a_{{3}}{a_{{4}}}^{2}b_{{0}}{b_{{1}}}^{2}b_{{2}}-3\,a_{ {1}}{a_{{4}}}^{3}{b_{{0}}}^{3}b_{{3}}+3\,a_{{1}}{a_{{4}}}^{3}{b_{{0}}} ^{2}b_{{1}}b_{{2}} \cr & -a_{{1}}{a_{{4}}}^{3}b_{{0}}{b_{{1}}}^{3}+{a_{{2}}}^{4}{b_{{0}}}^{2}{b _{{4}}}^{2}-{a_{{2}}}^{3}a_{{3}}{b_{{0}}}^{2}b_{{3}}b_{{4}}-2\,{a_{{2} }}^{3}a_{{4}}{b_{{0}}}^{2}b_{{2}}b_{{4}}+{a_{{2}}}^{3}a_{{4}}{b_{{0}}} ^{2}{b_{{3}}}^{2}+{a_{{2}}}^{2}{a_{{3}}}^{2}{b_{{0}}}^{2}b_{{2}}b_{{4} }+3\,{a_{{2}}}^{2}a_{{3}}a_{{4}}{b_{{0}}}^{2}b_{{1}}b_{{4}} \cr & -{a_{{2}}}^{2}a_{{3}}a_{{4}}{b_{{0}}}^{2}b_{{2}}b_{{3}}+2\,{a_{{2}}}^{ 2}{a_{{4}}}^{2}{b_{{0}}}^{3}b_{{4}}-2\,{a_{{2}}}^{2}{a_{{4}}}^{2}{b_{{0 }}}^{2}b_{{1}}b_{{3}}+{a_{{2}}}^{2}{a_{{4}}}^{2}{b_{{0}}}^{2}{b_{{2}}} ^{2}-a_{{2}}{a_{{3}}}^{3}{b_{{0}}}^{2}b_{{1}}b_{{4}}-4\,a_{{2}}{a_{{3} }}^{2}a_{{4}}{b_{{0}}}^{3}b_{{4}}+a_{{2}}{a_{{3}}}^{2}a_{{4}}{b_{{0}}} ^{2}b_{{1}}b_{{3}} \cr & +3\,a_{{2}}a_{{3}}{a_{{4}}}^{2}{b_{{0}}}^{3}b_{{3}}-a_{{2}}a_{{3}}{a_{{ 4}}}^{2}{b_{{0}}}^{2}b_{{1}}b_{{2}}-2\,a_{{2}}{a_{{4}}}^{3}{b_{{0}}}^{ 3}b_{{2}}+a_{{2}}{a_{{4}}}^{3}{b_{{0}}}^{2}{b_{{1}}}^{2}+{a_{{3}}}^{4} {b_{{0}}}^{3}b_{{4}}-{a_{{3}}}^{3}a_{{4}}{b_{{0}}}^{3}b_{{3}}+{a_{{3}} }^{2}{a_{{4}}}^{2}{b_{{0}}}^{3}b_{{2}} \cr & -a_{{3}}{a_{{4}}}^{3}{b_{{0}}}^{3}b_{{1}}+{a_{{4}}}^{4}{b_{{0}}}^{4} \cr}$$

Thank you very much! I now agree with dxiv that the general case is not nice.. Actually, I wanted to assume that $a_4=b_4=1$;I hope that it is slightly nicer in that case. — user237522, Jun 28 '18 at 03:48
Well, according to WA it is still not so nice if we further assume that $a_4=b_4=1$.. — user237522, Jun 28 '18 at 04:14
Arguably, less nice. Without that assumption, each term is homogeneous of order $4$ in the $a$'s and $4$ in the $b$'s. — Robert Israel, Jun 28 '18 at 05:02
Thanks for your comment; I like it (however, I had in mind "nicer" in the sense that the resultant can be expressed by $8$ "unknown" coefficients instead of by $10$ "unknown" coefficients). — user237522, Jun 28 '18 at 05:23
So just erase (i.e. substitute $1$ for) each occurrence of $a_4$ or $b_4$ in the general result. — Robert Israel, Jun 28 '18 at 07:17

Computing the resultant of two polynomials of degree $4$

1 Answers1