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I'm looking at how to transform a cubic into a Weierstrass equation on page 52 of this elliptic curves pdf here. The author writes: “There are two distinct transformations depending on whether the point $P$ is an inflection point on $C$ or not.” On page 53 he gives the point $P\left(1,1,1\right)$ on the cubic $$X^{3}+2Y^{3}-3Z^{3}=0$$as an example of a point that isn't an inflection point. Apart from doing the calculations and seeing that the tangent line at this point meets the curve again, how to tell that $P$ isn't an inflection point?

Peter4075
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  • The link is not working (as it is), a chrome-extension is part of the link declaration. What is wrong about doing the computations? For instance, write $X=1+x$, $Y=1+y$, $Z=1+z$, isolate the linear part in $(x,y,z)$... – dan_fulea Dec 08 '23 at 13:12
  • @dan_fulea - I've changed the link. Sorry, I'm a bit slow on the uptake. How does your suggestion help me show that P isn't an inflection point? You can't make your explanation too simple :-) – Peter4075 Dec 08 '23 at 13:59
  • We insert $X,Y,Z$ (well, the formulas break the homogenity, but it is ok for the purpose) in the equation and obtain something like$$(3x+6y-9z)+\text{(higher terms)}=0\ ,$$Taylor expansion around $P=(1,1,1)$ or $[1:1:1]$. The first derivative is alternative. The tangent line is the linear part, $0=3x+6y-9z$, so a point on this line is of the shape $x=-2y+3z$. We plug in this $x$ to obtain a polynomial in two variables, $y,z$, and factor,$$(1-2y+3z)^3+2(1+y)^3-3(1+z)^3\=(1-2y+3z)^3-(1+z)^3+2(1+y)^3-2(1+z)^3\=(y-z)(\dots)\ ,$$and it turns out that we finally factor $(y - z)^2(y - 4z - 3)$... – dan_fulea Dec 08 '23 at 14:45
  • But this is exactly the same story, building the tangent and looking for the intersection points.. – dan_fulea Dec 08 '23 at 14:46
  • @dan_fulea - Many thanks. I'd seen references daunting references online that I needed to use Hessian matrices to determine inflection points (they used the term "saddle point"). I've never studied Hessian matrices, so that confused me. So I can ignore Hessian matrices and just do the calculation, either as you do, or (which I'm more familiar with) by finding the tangent at P and looking for the intersection points? If there's only one such point, P is an inflection point. Is that correct? – Peter4075 Dec 08 '23 at 15:22

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