I am working with a dot product of 2 unit vectors in R3 that are algebraic.
I am trying to recover their original format in a number field, but I am not sure of how to go about doing this. I do use GP-Pari, but apparently it has come time to learn simple things about number fields.
Let one vector in R[3] be called A, its coefficients[x,y,z] are from roots of integer coefficient polynomials, (using GP-Pari) where poly[root] is the root # of the polynomial roots as listed out by GP-Pari command polroots().
A[1] = polroots(98*x^12 - 2814*x^10 - 9555*x^9 + 25335*x^8 + 133140*x^7 + 146288*x^6 - 278778*x^5 + 3780*x^4 + 64836*x^3 - 6090*x^2 - 2571*x + 59)[2]
A[2] = polroots(9604*x^24 + 436296*x^22 + 7451136*x^20 + 104590437*x^18 + 1010241768*x^16 + 2770851456*x^14 + 38854511616*x^12 - 268919935488*x^10 + 614263209984*x^8 - 702584930304*x^6 + 438328295424*x^4 - 142973337600*x^2 + 19138609152)[11]
A[3] = 0
which is all well and good, since I know what the values are. It is the second vector whose components I wish to recover
B[1] = polroots(98*x^12 - 2814*x^10 - 9555*x^9 + 25335*x^8 + 133140*x^7 + 146288*x^6 - 278778*x^5 + 3780*x^4 + 64836*x^3 - 6090*x^2 - 2571*x + 59)[5]
B[2] = ? (suspected 36th power or 72nd power field? or 144th power? unknown)
B[3] = ? (unknown)
Here is what is known:
A dot B = polroots(14*x^9 + 42*x^8 + 804*x^7 + 935*x^6 + 9318*x^5 - 6627*x^4 - 13488*x^3 + 8217*x^2 + 4536*x - 2727)[3]
A[1] * B[1] = polroots(941192*x^18 + 27025656*x^17 + 137462052*x^16 - 6953962106*x^15 - 84970588458*x^14 + 117612302514*x^13 + 8850508638259*x^12 + 7009827208866*x^11 - 62417542754208*x^10 - 174929280314834*x^9 - 5794674063903*x^8 + 222618099273210*x^7 + 228018482057086*x^6 + 48834186405150*x^5 - 686321579547*x^4 - 399708345506*x^3 - 118507164*x^2 + 57123210*x + 205379)[2]
A[2] * B[2] = A dot B - A[1]*B[1] = ?
A[3] * B[3] = 0 since A[3] = 0
But we also know that B[1]^2 + B[2]^2 + B[3]^2 = 1, since A & B are unit vectors, so recovering B[2] should enable us also to find B[3]. (B[1] is known)
I seek to find B[2] and B[3] from the information above. I have carefully checked these polynomial equations and I believe that they are correct.
How can this be done in a number field? I understand that I am decomposing the dot product A dot B = A[1]*B[1] + A[2]*B[2] + A[3]*B[3] = A[1]*B[1] + A[2]*B[2] + 0 and eventually getting around to finding a quotient (A dot B - A[1]*B[1]) / A[2] = B[2] in some type of number field in order to obtain B[2]. Then I have to find some square root in the number field, since 1 = B[1]^2 + B[2]^2 + B[3]^2 in order to find B[3].
I would appreciate some tips on using the number field functions of GP Pari to determine the answer, as I have other vectors with unknown algebraic components that I wish to determine, and it takes a similar process, as the one given above, in order to obtain their components. (I only know 1 out of 3, the other 2 are unknown)
Thanks for showing how to use the GP-Pari commands to solve this interesting problem.
