June 7, 2001 Try to solve the following maths problem: does x^3+y^2+1 produce the same form as x^3+3y^2+xy^2? For cubic equations, it’s possible to solve this problem, but mathematicians found things more difficult in the case of equations of the fourth order such as x^4+y^3+2y^2=3, a curve shaped like a lampshade. Eindhoven NWO researchers, funded by NWO Exact Sciences, have now found a method of dealing with equations of this type.
As part of an NWO project at Eindhoven University of Technology (TUE), mathematicians considered whether two different equations produced the same graph. In other words, are the two equations equivalent? They managed to solve this problem for fourth-order equations. The method also works for more complex fifth-order equations. The team first converted the fourth-order equations into quadratic equations, using so-called covariant mapping. Information is lost by doing this, but the resulting quadratic equations are easy to comprehend and to solve.
The team then used the solution to the quadratic equations to cancel out part of the original fourth-order equations. The remaining equations can then be solved with a bit of calculation. This new method can be used for equations of the fourth order and –with some modifications– for those of the fifth order. The Eindhoven team intend going on to see whether sixth-order or even more complex equations can be solved in the same way.
The problem of equivalence plays a role in such things as computer recognition of images. A computer is not able to see, for example, that an actual lampshade shown in two different ways is in fact the same lampshade. If the computer can convert the two images into mathematical formulae, it can use the new calculation method to work out that they represent the same lampshade. However, that is still in the future. Computers find the first step difficult, namely converting the lampshade into a mathematical formula.
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