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[函数] 齐次多项式 极化恒等式

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hbghlyj posted 2025-6-23 15:29 |Read mode
ncatlab.org/nlab/show/polarization+identity#higher_order
设 Q 为 p 次齐次多项式,其对称化为 m
  • If $p = 0$, $m() = Q(0)$
  • If $p = 1$, $m(v_1) = Q(v_1)$
  • If $p = 2$, $m(v_1, v_2) = Q(v_1 + v_2) - Q(v_1) - Q(v_2)$
  • If $p = 3$, $m(v_1, v_2, v_3) = Q(v_1 + v_2 + v_3) - Q(v_1 + v_2) - Q(v_1 + v_3) - Q(v_2 + v_3) + Q(v_1) + Q(v_2) + Q(v_3)$
  • If $p = 4$, $m(v_1, v_2, v_3, v_4) = Q(v_1 + v_2 + v_3 + v_4) - Q(v_1 + v_2 + v_3) - Q(v_1 + v_2 + v_4) - Q(v_1 + v_3 + v_4) - Q(v_2 + v_3 + v_4)$$+ Q(v_1 + v_2) + Q(v_1 + v_3) + Q(v_1 + v_4) + Q(v_2 + v_3) + Q(v_2 + v_4) + Q(v_3 + v_4) - Q(v_1) - Q(v_2) - Q(v_3) - Q(v_4)$

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