A curve that intersects $2x^3y^3-30x^2y^2+11xy^3+2x^3-38x^2y+20xy^2-13y^3+16x^2+94xy+10y^2+301x-668y+662$ with multiplicity $3$ or more

Question

let $C=2x^3y^3-30x^2y^2+11xy^3+2x^3-38x^2y+20xy^2-13y^3+16x^2+94xy+10y^2+301x-668y+662$ be a curve in $\mathbb{C^2}$. Find a conic that intersects $C$ in $(2,2)$ with intersection multiplicity $\geq 3$.

Since $C_x(2,2) \neq 0$ and $C_y(2,2) \neq 0$ it is a nonsingular point for $C$. We have defined the multiplicity of intersection of two curves in the projective plane so I imagine I need to find a conic $D=a_{00}x^2+2a_{01}xy+2a_{02}xz+a_{11}y^2+2a_{12}yz+a_{22}z^2$ such that eliminating $z$ from $D$ and $C^h$, where $C^h$ is the homogenization of $C$, the factor $(x-y)^3$ divides such polynomial.

I computed such conditions but they are quite unusable in this form so I opted for a trial-and-error approach, but all of the conics I tried have i.m. $2$, like the circle $(x-\frac{7}{4})^2+(y-3)^2=\frac{17}{16}$.

Any idea or hint? Thank you.

Answer 1

If it is not required the conic should be smooth, you could take the union of the tangent to $C$ at $(2,2)$ and any other line that passes through $(2,2)$.

Answer 2

Writing your equation about $(2,2)$ as $$2(x-2)^3(y-2)^3+12(x-2)^2(y-2)^3+35(x-2)(y-2)^3+25(y-2)^3+12(x-2)^3(y-2)^2+42(x-2)^2(y-2)^2+110(x-2)(y-2)^2+80(y-2)^2+24(x-2)^3(y-2)-14(x-2)^2(y-2)-38(x-2)(y-2)-612(y-2)+18(x-2)^3-72(x-2)^2+153(x-2)=0,$$ (noting the "constant term" is $0$ since $(2,2)$ is on the curve). Now keeping the terms of degree less than $2$: $$80(y-2)^2-38(x-2)(y-2)-72(x-2)^2+153(x-2)-612(y-2)=0$$ you get the conic fit (the red hyperbola in the plot).

Now the intersection multiplicity of the curve with the conic fit is $3$ at $(2,2)$, since by construction the first difference is in the "homogeneous component" of degree $3.$

This is analogous with the tangent line having intersection multiplicity $2$ when we do the same process with just the tangent or linear "homogeneous component" $$153(x-2)-612(y-2)=0.$$

Edit: But we can do better! The osculating conic at the point $(2,2)$ of the curve has contact $\geq 5$ and is computable through a puiseux expansion there. In Magma

A := AlgebraicClosure();
S<y> := PuiseuxSeriesRing(A);
P<x> := PolynomialRing(S);
f := 2*x^3*y^3+12*x^2*y^3+35*x*y^3+25*y^3+12*x^3*y^2+42*x^2*y^2+110*x*y^2+80*y^2+24*x^3*y-14*x^2*y-38*x*y-612*y+18*x^3-72*x^2+153*x;
time S := PuiseuxExpansion(f, 4);
A;
S

Time: 0.010
Algebraically closed field with 2 variables over Rational Field
Defining relations:
[
    r2^2 - 4*r2 + 17/2,
    r1^2 - 4*r1 + 17/2
]
[
    4*y + 8*y^2 + 23*y^3 + 62*y^4 + O(y^5),
    r1 + (-37/27*r1 - 83/54)*y + (277/324*r1 - 419/81)*y^2 + (1679/972*r1 -
    3574/243)*y^3 + O(y^4),
    r2 + (-37/27*r2 - 83/54)*y + (277/324*r2 - 419/81)*y^2 + (1679/972*r2 -
    3574/243)*y^3 + O(y^4)
]

$(x,y)=(4t + 8t^2 + 23t^3 + 62t^4+\ldots,t)$

which gives a general fit (available on request; the idea is this answer) to the puiseux expansion which at $t=0$ and shifted back to the point becomes

$-33(x-2)^2+1736(x-2)(y-2)-2320(y-2)^2-512(x-2)+2048(y-2)=0$

which you can check has intersection multiplicity $5$ at $(2,2)$.