Milnor-Thom theorem

In the mathematical subfield of algebraic geometry gives set of Milnor-Thom an estimate for the number of connected components of the set of zeros of a polynomial , and more generally for the sum of the Betti numbers of zero set.

Sets of zeros of polynomials

Let it be a polynomial in variables of degree . Milnor-Thom's theorem gives estimates for the topology of the root set ${\displaystyle P\in \mathbb {R} \left[x_{1},\ldots ,x_{n}\right]}$ ${\displaystyle n}$ ${\displaystyle k}$

${\displaystyle A=\left\{x\in \mathbb {R} ^{n}:P(x)=0\right\}\subset \mathbb {R} ^{n}}$,

more precisely for the sum of the Betti numbers . ${\displaystyle b_{i}(A)}$

Because the number of connected components of th 0 equal to the Betti number is and all Betti numbers not negatively are obviously applies ${\displaystyle A}$ ${\displaystyle b_{0}(A)}$

${\displaystyle \sharp \left\{{\text{Zusammenhangskomponenten von }}A\right\}=b_{0}(A)\leq b_{0}(A)+b_{1}(A)+\ldots +b_{n}(A)}$

and in particular an estimate for the number of connected components is obtained from Milnor-Thom's theorem.

The inequalities

John Milnor , in his 1964 paper, looked more generally at algebraic varieties defined by polynomials , each of degree, and proved that the sum of their Betti numbers was the inequality ${\displaystyle V\subset \mathbb {R} ^{n}}$ ${\displaystyle p}$ ${\displaystyle f_{i}=0,i=1,\ldots ,p,}$ ${\displaystyle \leq k}$

${\displaystyle b_{0}(V)+b_{1}(V)+\ldots +b_{n}(V)\leq k(2k-1)^{n-1}}$

Fulfills. For the case that is defined by polynomial inequalities , he proved ${\displaystyle V}$ ${\displaystyle f_{i}\geq 0,i=1,\ldots ,p}$

${\displaystyle b_{0}(V)+b_{1}(V)+\ldots +b_{n}(V)\leq {\frac {1}{2}}(2+d)(1+d)^{n-1}}$

with . He also proved inequalities for complex algebraic varieties and for projective varieties . ${\displaystyle d=deg(f_{1})+\ldots +deg(f_{p})}$ ${\displaystyle V\subset \mathbb {C} ^{n}}$

René Thom , in his work published in 1965 but already written earlier, proved the estimate for the set of zeros of a polynomial of degree . Both proofs, from Milnor and from Thom, used Morse theory . ${\displaystyle A\subset \mathbb {R} ^{n}}$ ${\displaystyle k}$ ${\displaystyle b_{0}(A)+b_{1}(A)+\ldots +b_{n}(A)\leq (2k)^{n}}$

In 1996, Nolan Wallach gave an improved estimate for the case of non-singular hypersurfaces: if a polynomial is of degree and a regular value of , then the inequality holds for the sum of the Betti numbers of ${\displaystyle f\colon \mathbb {R} ^{n}\to \mathbb {R} }$ ${\displaystyle k}$ ${\displaystyle 0}$ ${\displaystyle f}$ ${\displaystyle A=f^{-1}(0)}$

${\displaystyle b_{0}(A)+b_{1}(A)+\ldots +b_{n}(A)\leq k^{n}}$.

literature

• Thom: Sur l'homologie des variétés algébriques réelles. Differential and Combinatorial Topology (A Symposium in Honor of Marston Morse) pp. 255-265 Princeton Univ. Press, Princeton, NJ (1965) Online
• Milnor: On the Betti numbers of real varieties. Proc. Amer. Math. Soc. 15: 275-280 (1964), JSTOR 2034050 .
• Wallach: On a theorem of Milnor and Thom in: Topics in Geometry (Simon Gindikin, editor), 331-348, Progr. Nonlinear Differential Equations Appl., 20, Birkhauser Boston, Boston, MA, 1996. Online MR1390322
• Jacek Bochnak, Michel Coste, Marie-Françoise Roy : Real Algebraic Geometry, Springer 1998, Chapter 11.5 (The sentence is on p. 284)