# Projekt:Computeralgebra-Berechnungen/Symmetrische Hilbert-Kunz Theorie/Syz2/Fermat-Quartik (vier Variaben)/Maximales Ideal

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Zum maximalen Ideal ${\displaystyle {}(x,y,z,w)}$ auf der Fermat-Quartik ${\displaystyle {}x^{4}+y^{4}+z^{4}+w^{4}=0}$ werden die symmetrische Asymptotik und die Frobenius-Asymptotik des zweiten Syzygienbündels verglichen. Aus der bekannten Frobenius-Hilbert-Kunz Multiplizität kann man errechnen, dass der ${\displaystyle {}H^{2}}$-Frobenius Grenzwert rechts gleich ${\displaystyle {}18,6666}$ ist.

q ${\displaystyle {}\sum h^{2}(S^{q}(Syz_{2})(m))}$ durch Rang durch Rang ${\displaystyle {}q^{3}}$ 1 141 35,25 35,25 ${\displaystyle {}-}$ ${\displaystyle {}-}$ ${\displaystyle {}-}$ 2 1961 196,1 24,5125 ${\displaystyle {}-}$ ${\displaystyle {}-}$ ${\displaystyle {}-}$ 3 11841 592,05 21,9277 2437 609,25 22,5648 4 46610 1331,7142 20,8080 ${\displaystyle {}-}$ ${\displaystyle {}-}$ ${\displaystyle {}-}$ 5 141147 (in ${\displaystyle {}\mathbb {Z} /(7)}$) 2520,4821 20,1638 10383 2595,75 20,766 6 ${\displaystyle {}-}$ ${\displaystyle {}-}$ ${\displaystyle {}-}$ 7 27591 6897,75 20,1100 8 9 10 ${\displaystyle {}-}$ ${\displaystyle {}-}$ ${\displaystyle {}-}$ 11 104073 26018,25 19,5478 12 ${\displaystyle {}-}$ ${\displaystyle {}-}$ ${\displaystyle {}-}$ 13 170515 42628,75 19,4031 14 ${\displaystyle {}-}$ ${\displaystyle {}-}$ ${\displaystyle {}-}$ 15 ${\displaystyle {}-}$ ${\displaystyle {}-}$ ${\displaystyle {}-}$ 16 17 377749 94437,25 19,2219 18 ${\displaystyle {}-}$ ${\displaystyle {}-}$ ${\displaystyle {}-}$ 19 525709 131427,25 19,1612 20 ${\displaystyle {}-}$ ${\displaystyle {}-}$ ${\displaystyle {}-}$