line at infinity

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line at infinity

[′līn at in′fin·əd·ē]
(mathematics)
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* 1) How many distinct configurations of invariant lines, including the line at infinity, of total multiplicity 6 (respectively 5, 4) of non-degenerate systems in QS with a finite number of singularities at infinity do we have?
Systems in this class which possess invariant lines of total multiplicity 6, including the line at infinity, have a total of II possible configurations of invariant straight lines.
2) If the line at infinity Z = 0 is filled up with singularities, in each one of the charts at infinity X [not equal to] 0 and Y [not equal to] 0, the corresponding system in the Poincare compactification is degenerate and we need to do a rescaling of an appropriate degree of the system, so that the degeneracy be removed.
Hence these are also results of total multiplicity 8 if we include in our counting the line at infinity. The problem 3) is not completely solved.
3) We have a total of 11 topologically distinct phase portraits for systems in QS with the line at infinity filled up with singularities for a total of 9 distinct configurations of affine invariant lines.
4) There are exactly 112 topologically distinct Lotka-Volterra systems: 60 of them with exactly three invariant lines, all simple; 27 portraits with invariant lines with total multiplicity at least four; 5 with the line at infinity filled up with singularities; 20 phase portraits of degenerate systems.

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