A *conic* in the plane $\mathbb{R}^2$ is the zero set of a
quadratic polynomial in two variables:

$$ \,\, a_1 x^2 \,+\, a_2 xy \,+\, a_3 y^2 \, +\, a_4 x \, + \, a_5 y \, + \, a_6 \,.$$

Geometrically, a conic can be either a circle, an ellipse, a hyperbola, a parabola or a union of two lines. The last case is a degenerate conic. Double lines are allowed.

A classic problem in enumerative geometry is Steiner’s conic problem which asks:

*How many conics are tangent to five given conics?*

Here is an example of Steiner’s problem:

Steiner claimed in 1848 that there are at most 7776 conics tangent to 5 given conics.
He phrased his problem as that of solving five equations
of degree six on the 5-dimensional projective space $\mathbb{P}^5$.
Using Bézout’s Theorem he argued that the equations coming from this question
have $6^5 = 7776$ solutions over the complex numbers.
However, this number over counts because
there is a *Veronese surface* of extraneous solutions, namely
the conics that are squares of linear forms, i.e., double lines.

The correct count of non-degenerate conics is **3264**.
This was shown in 1859 by Jonquières and
independently in 1864 by Chasles.
The number 3264 appears prominently in the book
*3264 and all that*
by Eisenbud and
Harris.

A delightful introduction to Steiner’s problem was given by
Bashelor, Ksir and Traves.
Ronga, Tognoli and Vust and Sottile showed how to
choose 5 real conics such that all
3264 complex solutions are *real*.
Although their proofs starts with an explicit construction, it is *not* constructive.

Using methods from numerical algebraic geometry
we adapted the proposed construction to find an explicit instance for which there are 3264 real conics.
We use **numerical homotopy continuation** to compute the 3264 conics.
Smale’s $\alpha$-theory provides a way to give a **mathematical proof** that we found 3264 real conics; the keyword is alphaCertified.
The computational proof can be downloaded here.

An illustration of the arrangement is shown below.

It looks like the arrangement consists of 5 blue lines, rather than 5 blue conics. The next picture clarifies the situation.

More details on the geometry of these five special conics are in our article 3264 conics in a Second. There is also an interactive plot for GeoGebra, which lets you zoom in at the points of tangency.