By Eduard L. Stiefel

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I=0 i 1. Prove that F 2 (x) = ∑∞ i=0 Ci+1 x = F(x)−1 . 4. The rest of this book 39 √ 1 + 1 − 4x . ) 2. Deduce that x→0+ 3. Recall Newton’s binomial theorem: (1 + z)1/2 = ∞ ∑ k=0 1/2 k z, k where the fractional binomial is defined as: 1/2 1/2(1/2 − 1)(1/2 − 2) . (1/2 − k + 1) . = k! k Apply it to z = −4x and express the fractional binomial in terms of 2k k . 6. 3) holds. 7. 3 is equivalent to the following one: Given a triangulation of the (n + 2)-gon, the sequence consists of n + 1 blocks, one for each vertex of the polygon other than the first one.

The first part is particularly important. 43: Eight configurations are possible in degree 5. Only the two maximal ones are shown. 28 Triangulations in Mathematics degree consists just in telling how many different ovals can appear and what the possible nesting structures are. In other words, we want to know which partially ordered sets (posets) can arise in this way from a smooth curve of given degree. The equivalence class of curves that induce the same poset is an isotopy class. 8 (Harnack). The number of connected components of a nonsingular algebraic curve f of degree d is at most (d 2 − 3d + 4)/2, or, equivalently, one plus the genus of the Riemann surface associated with f .

As usual, C∗ denotes C \ {0}. That is, the roots to which the theorem applies are those with no zero coordinate. In algebraic geometry, (C∗ )d is usually called the complex torus of dimension d. As an application of Bernstein’s theorem, consider the system of equations ax3 y2 + bx + cy2 + d = 0 and exy4 + f x3 + gy = 0. The Bézout bound estimates 25 complex roots. The number of roots in the torus (C∗ )2 predicted by Bernstein’s Theorem is 18. Using Gröbner bases one can see that 18 is in fact the actual number of roots in C2 .

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An Introduction to Numerical Mathematics by Eduard L. Stiefel

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