Misfortunes of a mathematicians' trio using Computer Algebra Systems: Can we trust?

Computer algebra systems are a great help for mathematical research but sometimes unexpected errors in the software can also badly affect it. As an example, we show how we have detected an error of Mathematica computing determinants of matrices of integer numbers: not only it computes the determinants wrongly, but also it produces different results if one evaluates the same determinant twice. MSC Numbers: 68W30 Introduction Nowadays mathematicians often use computer algebra systems as an assistant in their mathematical research. Mathematicians have the ideas, and tedious computations are left to the computer. Everybody “knows” that computers perform this work better than persons. But, of course, we must trust in the results derived by the powerful computer algebra systems that we use. Currently we are using Mathematica to find examples and counterexamples of some mathematical results that we are working out, with the aim of finding the correct hypothesis and later to build a mathematical proof. Our goal was to improve some results by Karlin and Szegő [4] related to orthogonal polynomials on the real line. Details are not important, and this is just an example of the use of a computer algebra system by a typical mathematician in research, but let us explain it briefly; it is not necessary to completely understand it, just to see that it was a typical mathematical research with computer algebra as a tool. Our starting point is a discrete positive measure on the real line μ = ∑ n≥0 Mnδan (where δa denotes a Dirac delta in a, and an < an+1), having a sequence of orthogonal polynomials {Pn}n≥0 (where Pn has degree n and positive leading coefficient). Karlin and Szegő considered in 1961 (see [4]) the l × l Casorati determinants det 