Chapter 5: Two Dimensions and Beyond

Section 7: Systems Based on Constraints

Constraints on formulas

Many standard problems of algebraic computation can be viewed as consisting in finding formulas that satisfy certain constraints. An example is exact solution of algebraic equations. For quadratic equations the standard formula gives solutions for arbitrary coefficients in terms of square roots. Similar formulas in terms of n^th roots have been known since the 1500s for equations with degrees n up to 4, although their LeafCount starting at n=1 increases like 6, 25, 183, 718. For higher degrees it is known that such general formulas must involve other functions. For degrees 5 and 6 it was shown in the late 1800s that EllipticTheta or Hypergeometric2F1 are sufficient, although for degrees 5 and 6 respectively the necessary formulas have a LeafCount in the billions. (Sharing common subexpressions yields a LeafCount in the thousands.) (See also page 1129.)

From Stephen Wolfram: A New Kind of Science [citation]