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On September 30, 1744, the Swiss mathematician Gabriel Cramer (1704鈥�1752) wrote a remarkable letter to his countryman Leonhard Euler, concerning the issue of fitting a cubic to given points in the plane. He states two 鈥渇acts鈥� about cubics: (1) Any nine distinct points determine a unique cubic. (2) Two cubics can intersect in nine points. Cramer points out that these two statements are incompatible. If we consider two specific cubics that intersect in nine points (suchas$x^3鈭�x=0$and), then there is more than one cubic through these nine points, contradicting the first 鈥渇act.鈥� Something is terribly wrong here, and Cramer asks Euler, the greatest mathematician of that time, to resolve this apparent contradiction. (This issue is now known as the Cramer鈥揈uler paradox.)

Euler worked on the problem for a while and put his answer into an article he submitted in 1747, 鈥淪ur one contradiction apparentedans la doctrine des lignescourbes鈥� [Memoires de l鈥橝cad 麓 emie des Sciences de 麓 Berlin, 4 (1750): 219鈥�233].

Using Exercises 46 through 59 as a guide, explain which of the so-called facts stated by Cramer is wrong, thus resolving the paradox.

Step by step solution

01

Describe the given information

The given facts are,

1. Any nine distinct points determine a unique cubic.

2. Two cubics can intersect in nine points

02

Explain which of the so-called facts stated by Cramer is wrong

Cramer鈥檚 second statement is true. For instance, the cubic $x(x鈭�1)(x鈭�2)$and the cubic$y(y鈭�1)(y鈭�2)$intersect at the following nine points.

$\{(0,0),\text{鈥�}(0,1),\text{鈥�}(0,2),\text{鈥�}(1,0),\text{鈥�}(1,1),\text{鈥�}(1,2),\text{鈥�}(2,0),\text{鈥�}(2,1),\text{鈥�}(2,2)\}$

However, Cramer鈥檚 first statement is not always true.

Consider exercise 53.

In this case, the nine points defined a single cubic. However, for exercise 48 and 54, the nine points defined a family of cubic.

Observed that the criterion for the nine points to define a single cubic is that the matrix$A$should have rank $9$, as in exercise 53. In exercises 48 and 54, the matrix had rank$8$and the cubic was not uniquely defined.

The matrix is defined by,

$A=\left[\begin{array}{cccccccccc}1& x_1& y_1& x_1^2& x_1{y}_1& y_1^2& x_1^3& x_1^2{y}_1& x_1{y}_1^2& y_1^3\\ 1& x_2& y_2& x_2^2& x_2{y}_2& y_2^2& x_2^3& x_2^2{y}_2& x_2{y}_2^2& y_2^3\\ 1& x_3& y_3& x_3^2& x_3{y}_3& y_3^2& x_3^3& x_3^2{y}_3& x_3{y}_3^2& y_3^3\\ 鈰�& 鈰�& 鈰�& 鈰�& 鈰�& 鈰�& 鈰�& 鈰�& 鈰�& 鈰�\\ 1& x_9& y_9& x_9^2& x_9{y}_9& y_9^2& x_9^3& x_9^2{y}_9& x_9{y}_9^2& y_9^3\end{array}\right]$

Hence, the Cramer鈥檚 second statement is true, and Cramer鈥檚 first statement is not always true.

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