Download pdf - In[1]:= dim H This is dimension we are intersted in L D DD H D L

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  • In[1]:= dim = 6 H* This is dimension we are intersted in *Lxx = Append@Array@x, dimD, x@0DDH* A set of inderetminates x@0D - homogenisation variable*L

  • In[11]:= rr = quadRRule@xx, cD; H* replacement rule for squared homogenised variables x *Lrri = Flatten@

    Table@If@Length@Variables@rr@@4DD@@kDDDD > 0, rr@@4DD@@kDD ® rr@@3DD@@kDD , 8

  • In[29]:= 2023 - 1959 H* rank deficit,since we are inposing F_2 on 6 variables the expected max value is 2^6= 64 *L

    Out[29]= 64

    In[30]:= H* generate all 2^6 possible values of x to test they are in the null space *LxVal = Flatten@Table@8k1, k2, k3, k4, k5, k6, 2< * 2 - 3,

    8k1, 2

  • In[35]:= fx = Table@xx@@kDD^2 - x@0D^2, 8k, dim

  • --------------- Forth order -------------------

    Rank deficency

    57

    Norm of values

    0

    Rank of values

    57

    ---------------- Fifth order ------------------

    Rank deficency

    63

    Norm of values

    0

    Rank of values

    63

    ----------------- Sixth order -----------------

    Rank deficency

    64

    Norm of values

    0

    Rank of values

    64

    and now certificates for factoring first in ususal Nulstellensatz linear algebra

    In[43]:= ff = Expand@HSum@Hxx@@kDD + x@0DL 2^Hk - 1L, 8k, dim 2< D + x@0DLHSum@Hxx@@k + dim 2DD + x@0DL 2^Hk - 1L, 8k, dim 2< D + x@0DLD - 0

    Sort@Table@ ff . Table@xx@@kDD ® xVal@@mDD@@kDD, 8k, dim + 1

  • In[46]:= certEval@n_D := Block@ 8qM, fAll, cert, certRules, coefCond, ccca

  • Out[56]=

    A very large output was generated. Here is a sample of it:

    91, c@21D, c@21D2, c@21D3, c@20D, c@20D c@21D, c@20D c@21D2,c@20D2, c@20D2 c@21D, c@20D3, c@19D, c@19D c@21D, c@19D c@21D2,c@19D c@20D, c@19D c@20D c@21D, 1995, c@1D2 c@14D, c@1D2 c@13D,c@1D2 c@12D, c@1D2 c@11D, c@1D2 c@10D, c@1D2 c@9D, c@1D2 c@8D, c@1D2 c@7D,c@1D2 c@6D, c@1D2 c@5D, c@1D2 c@4D, c@1D2 c@3D, c@1D2 c@2D, c@1D3=

    Show Less Show More Show Full Output Set Size Limit...

    Out[57]= 4116

    Out[58]=

    A very large output was generated. Here is a sample of it:

    c@1D z@1D - c@1D3 z@1D + c@2D z@2D - c@1D2 c@2D z@2D + c@1D z@3D -c@1D c@2D2 z@3D + c@2D z@4D - c@2D3 z@4D - c@1D2 c@2D z@5D + c@1D c@3D z@5D +12 318 + c@19D z@4112D - c@19D c@21D2 z@4112D + c@14D c@21D z@4113D -c@19D c@21D2 z@4113D + c@20D z@4114D - c@20D c@21D2 z@4114D +c@15D c@21D z@4115D - c@20D c@21D2 z@4115D + c@21D z@4116D - c@21D3 z@4116D

    Show Less Show More Show Full Output Set Size Limit...

    Out[59]= 8c@1D, c@2D, c@4D, c@7D, c@8D, c@9D, c@10D,c@11D, c@12D, c@13D, c@14D, c@16D, c@17D, c@18D, c@19D<

    Out[61]=

    A very large output was generated. Here is a sample of it:

    883, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0< ® -z@1D,82, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0< ® -z@2D - z@5D,2021, 80, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0< ® 1<

    Show Less Show More Show Full Output Set Size Limit...

    Out[62]= 2024

    Out[63]=

    A very large output was generated. Here is a sample of it:

    8-z@1D, -z@2D - z@5D, -z@6D + z@7D, -z@19D - z@29D, -z@30D + z@31D,2015, -z@4116D, z@4061D + z@4062D + z@4063D + z@4064D + z@4065D,z@3206D + z@3218D + z@3237D + z@3257D + z@3284D + z@3320D + z@3352D +

    z@3391D + z@3439D + z@3498D + z@3536D + z@3578D + z@3627D + z@3685D +z@3754D + z@3802D + z@3854D + z@3913D + z@3981D + z@4060D + z@4116D, 1<

    Show Less Show More Show Full Output Set Size Limit...

    Out[64]= 8SparseArray@, 82024

  • In[67]:= 2024 - 1959

    Out[67]= 65

    In[240]:= ffr2 =

    Flatten@Table@ Expand@Hffr@@kDD + ffr@@mDDL^3 - Hffr - ffr@@kDD - ffr@@mDDL^3 D,8k, Length@ffrD - 1

  • Out[249]=

    A very large output was generated. Here is a sample of it:

    8512 t@1D - 512 t@2D - 512 t@3D - 512 t@4D - 512 t@5D - 512 t@6D -512 t@7D - 512 t@8D - 512 t@9D - 512 t@10D - 512 t@11D - 512 t@12D -512 t@13D - 512 t@14D - 512 t@15D + 512 t@16D + 195 + 8 y@253D - z@1D,

    182 + 8 y@252D + 16 y@253D - z@2D - z@5D, 2020,64 y@2D + z@3206D + z@3218D + 17 + z@4060D + z@4116D, 1 + 262 144 t@1D + 262 144 t@2D +

    262 144 t@3D + 262 144 t@4D + 262 144 t@5D + 262 144 t@6D + 219 + 64 y@1D<

    Show Less Show More Show Full Output Set Size Limit...

    Out[250]=

    A very large output was generated. Here is a sample of it:

    8t@1D, t@2D, t@3D, t@4D, t@5D, t@6D, t@7D, t@8D, t@9D, t@10D, t@11D, t@12D, t@13D,t@14D, t@15D, t@16D, t@17D, t@18D, t@19D, t@20D, t@21D, t@22D, t@23D, t@24D,t@25D, 4439, z@4092D, z@4093D, z@4094D, z@4095D, z@4096D, z@4097D, z@4098D,z@4099D, z@4100D, z@4101D, z@4102D, z@4103D, z@4104D, z@4105D, z@4106D, z@4107D,z@4108D, z@4109D, z@4110D, z@4111D, z@4112D, z@4113D, z@4114D, z@4115D, z@4116D<

    Show Less Show More Show Full Output Set Size Limit...

    Out[251]= 8SparseArray@, 82024

  • Out[252]=

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    10 tautologies6vars4thOrder.nb

  • Out[252]=

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    tautologies6vars4thOrder.nb 11

  • Out[252]=

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    0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,

    0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,

    0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,

    0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,

    0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,

    0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,

    0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,

    0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0>

    In[212]:= MatrixRank@ ccca@@2DD D

    Out[212]= 120

    12 tautologies6vars4thOrder.nb

  • In[213]:= Length@ffr2D

    Out[213]= 120

    In[253]:= Length@lsD

    Out[253]= 4489

    tautologies6vars4thOrder.nb 13


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