- #1
aheight
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- TL;DR Summary
- I don't understand Wikipedia's definition of when a polynomial system has a finite or infinite solution. That is, I don't know what "pure power of a variable in the Grobner basis" means:
Wish to determine when a system of polynomials has an infinite number of solutions, that is, is not zero-dimensional. The Wikipedia article : System of polynomial equations states:
I interpret the quote to mean the system has an infinite number of solutions if the Grobner basis does not have a leading monomial which is a pure power of the variable.
For example, I have the system:
$$
\begin{aligned}
g(z,w)&=\left(-\frac{976 w^2}{53}+\frac{178 w z}{71}+\frac{27 w}{94}+\frac{323 z^2}{84}+\frac{271 z}{67}+\frac{199}{28}\right) \\
h(z,w)&= \left(\frac{72 w^2}{89}-\frac{502 w z}{97}-\frac{44 w}{57}+\frac{73 z^2}{5}-\frac{369 z}{73}+\frac{483}{95}\right)
\end{aligned}
$$
Which Mathematica's NSolve returns four solutions. And I can compute the Grobner basis with the code below. The Grobner basis is:
$$
\begin{aligned}
&1.15564\times 10^{38} w^4-2.74482\times 10^{37} w^3-8.43405\times 10^{37} w^2+1.37174\times 10^{37} w+1.9389\times 10^{37}\\
&4.72185\times 10^{45} w^3-7.68083\times 10^{45} w^2-1.1989\times 10^{45} w+1.74982\times 10^{45} z+2.44771\times 10^{45}
\end{aligned}
$$
I've not been able to fine a definition of "pure power of a variable of Grobner basis" and was hoping someone could explain this to me. Thanks.
I interpret the quote to mean the system has an infinite number of solutions if the Grobner basis does not have a leading monomial which is a pure power of the variable.
For example, I have the system:
$$
\begin{aligned}
g(z,w)&=\left(-\frac{976 w^2}{53}+\frac{178 w z}{71}+\frac{27 w}{94}+\frac{323 z^2}{84}+\frac{271 z}{67}+\frac{199}{28}\right) \\
h(z,w)&= \left(\frac{72 w^2}{89}-\frac{502 w z}{97}-\frac{44 w}{57}+\frac{73 z^2}{5}-\frac{369 z}{73}+\frac{483}{95}\right)
\end{aligned}
$$
Which Mathematica's NSolve returns four solutions. And I can compute the Grobner basis with the code below. The Grobner basis is:
$$
\begin{aligned}
&1.15564\times 10^{38} w^4-2.74482\times 10^{37} w^3-8.43405\times 10^{37} w^2+1.37174\times 10^{37} w+1.9389\times 10^{37}\\
&4.72185\times 10^{45} w^3-7.68083\times 10^{45} w^2-1.1989\times 10^{45} w+1.74982\times 10^{45} z+2.44771\times 10^{45}
\end{aligned}
$$
I've not been able to fine a definition of "pure power of a variable of Grobner basis" and was hoping someone could explain this to me. Thanks.
Mathematica:
In[346]:=
g0[{z_, w_}] =
199/28 + (27 w)/94 - (976 w^2)/53 + (271 z)/67 + (178 w z)/71 + (
323 z^2)/84;
h0[{z_, w_}] =
483/95 - (44 w)/57 + (72 w^2)/89 - (369 z)/73 - (502 w z)/97 + (
73 z^2)/5;
NSolve[{g0[{z, w}] == 0, h0[{z, w}] == 0}, {z, w}]
GroebnerBasis[{g0[{z, w}], h0[{z, w}]}, {z, w}] // N
Out[348]= {{z -> 0.309562 - 0.532126 I,
w -> 0.688127 - 0.178627 I}, {z -> 0.309562 + 0.532126 I,
w -> 0.688127 + 0.178627 I}, {z -> 0.0622197 - 0.609678 I,
w -> -0.56937 + 0.088145 I}, {z -> 0.0622197 + 0.609678 I,
w -> -0.56937 - 0.088145 I}}
Out[349]= {1.9389*10^37 + 1.37174*10^37 w - 8.43405*10^37 w^2 -
2.74482*10^37 w^3 + 1.15564*10^38 w^4,
2.44771*10^45 - 1.1989*10^45 w - 7.68083*10^45 w^2 +
4.72185*10^45 w^3 + 1.74982*10^45 z}