I am looking for a way to compare the degree of two polynomials via resultant; I.e., given two polynomials $f(x)=f_0+f_1 x^1+\cdots+f_d x^d$ and $g(x)=g_0+g_1x^1+\cdots+g_s x^s$ in $K[f_0,\cdots,f_d,g_0,\cdots,g_s,x]$, is it possible to find a polynomial $R$ in $K[f_0,\cdots,f_d,g_0,\cdots,g_s]$ s.t. $R=0$ iff $d \leq s$? (Perhaps something like $res_x(f,g)$...)
Comparing degree of two polynomials via resultant
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polynomials
resultant
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2Latex.....use it. – 2017-02-01
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0The quotient of the polynomial long division $f / (x\cdot g)$ is $0$ iff $d \le s$. But you should really add some context and background, since that's a rather unusual question. – 2017-02-02
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1Yes: take $R$ identically $0$ if $d\le s$, and take $R$ identically $1$ if $d>s$. I doubt this is a sastifying answer, but it fits with how you've phrased the question. You'd really have to consider a polynomial ring over $K$ in infinitely many variables $f_0,f_1,\dots,g_0,g_1,\dots$, not separately consider polynomial rings in fixed numbers of variables, to avoid this unsatisfying answer. – 2017-02-02