三元六次半正定条件

数学星空

若设\(d_1=x+y+z,d_2=xy+xz+yz,d_3=xyz\),且\(x\geqslant0,y\geqslant0,z\geqslant0\),试找到：

\(f(x,y,z)=k_0 d_1^6+k_1 d_2 d_1^4+k_2 d_3 d_1^3+k_3 d_2^2 d_1^2+k_4 d_1 d_2 d_3 +k_5 d_2^3+k_6 d_3^2 \geqslant  0\)

成立的条件，并指出\(f(x,y,z)\)取\(＝\)的条件？

数学星空

上面的问题来自于：

http://bbs.cnool.net/cthread-105590209.html

数学爱好者xzlbq给出了如下结果：

则有　\([k_0,k_1,k_2,k_3,k_4,k_5,k_6,k]\)

1.\([0, 0, -10, 3, 36, -10, -27, k]\)
2.\([0, 0, -4, 1, 10, -2, -9, k]\)
3.\([0, 0, -2, 1, 10, -4, -9, k]\)
4.\([0, 1, 7, -8, -27, 16, 27, k]\)
5.\([0, 2, 2, -11, 2, 12, 9, k]\)
6.\([0, 4, -26, -25, -80, 90, 45, 1/3]\)
7.\([0, 8, 28, -53, -88, 84, 117, 2]\)
8.\([0, 9, 23, -56, -59, 80, 99, 3]\)
9.\([0, 25, 103, -172, -347, 288, 423, 5/3]\)
10.\([0, 49, 239, -352, -851, 624, 963, 7/5]\)
11.\([0, 50, 146, -319, -414, 476, 621, 5/2]\)
12.\([1, -20, -174, 169, 810, -420, -810, 1/13]\)
13.\([1, -18, -124, 134, 572, -312, -585, 1/11]\)
14.\([1, -17, -102, 118, 468, -264, -486, 1/10]\)
15.\([1, -16, -82, 103, 374, -220, -396, 1/9]\)
16.\([1, -15, -64, 89, 290, -180, -315, 1/8]\)
17.\([1, -14, -48, 76, 216, -144, -243, 1/7]\)
18.\([1, -13, -34, 64, 152, -112, -180, 1/6]\)
19.\([1, -12, -22, 53, 98, -84, -126, 1/5]\)
20.\([1, -11, -12, 43, 54, -60, -81, 1/4]\)
21.\([1, -10, -28, 41, 108, -64, -108, 1/2]\)
22.\([1, -10, -4, 34, 20, -40, -45, 1/3]\)
23.\([1, -9, 2, 26, -4, -24, -18, 1/2]\)
24.\([1, -7, 8, 13, -22, -4, 9, k]\)
25.\([1, -18, -12, 57, 54, -66, -81, k_{00}]\)
26.\([8, -108, -326, 559, 1462, -1020, -1683, 2/13]\)
27.\([8, -68, 34, 179, -98, -140, -63, 2/3]\)
28.\([9, -132, -526, 761, 2378, -1508, -2610, 3/23]\)

且\(f(x,y,z)\) 取\(=\)仅当\(x=y,y=kz\)成立

其中第25项中的\(0 \leqslant k _{00}\leqslant 0.03921970753\)
即满足：\(62k^6+204k^5+210k^4+200k^3+30k^2+24k-1\leqslant 0\)

上面是否给出了所有成立条件？