# Multivariate polynomial data fitting formula

A short description：

consider $$f(x,y,z)=x^2+y^2-z^2$$

we can know that $$f(1,0,1)=0,f(0,1,1)=0$$ ,etc.

so we can get the roots of this polynomial.

Now if given the roots of polynomial, how to solve the most possibly polynomial?

I have some interdependent data.There is a multivariate polynomial relationship between known data.

like : $$\{a_i, b_i, c_i, d_i, e_i\} , 1 \le i \le 30$$.

I want to generate a polynomial like : f[a, b, c, d, e] == 0。 The highest degree of polynomials f is no more than 5.

date = {{1.7959, 4.05481, 0.557256, 0.748305, -1.03845}, {0.0858696,
1.07169, 0.705519, 1.32677, -0.855295}, {4.90172, 0.158416,
0.878733, 1.96608, 954.245}, {2.27961, 1.71061, 1.44171,
0.69401, -1.84901}, {0.256048, 4.24589, 0.652135,
0.314675, -2.26791}, {0.016098, 3.35303, 0.0255727,
0.930836, -0.0371771}, {4.55962, 2.4236, 1.92493, 1.51859,
0.626972}, {4.64569, 4.67848, 1.69821,
1.35355, -1.3203}, {1.21131, 3.76428, 0.680899,
0.181542, -4.12665}, {3.52003, 4.06811, 0.153511,
1.118, -0.0969514}, {0.873221, 3.97128, 1.08425,
1.69994, -0.934625}, {2.06676, 3.31185, 1.81121,
1.937, -1.28445}, {3.23805, 0.966946, 0.420946, 0.341141,
7.18288}, {2.38365, 3.96166, 1.56003,
0.846376, -2.30139}, {2.70938, 0.336656, 0.0891704, 1.18989,
62.8609}, {4.29624, 0.725231, 0.313328, 1.99668,
33.2428}, {4.69485, 3.30499, 0.440388, 1.83629,
0.779267}, {1.98956, 3.55388, 0.252933,
1.08482, -0.403182}, {2.01274, 4.62648, 1.06508,
1.7516, -0.901361}, {2.68855, 0.445539, 1.045, 1.38863,
33.6256}, {0.758261, 3.13808, 1.90326,
0.687971, -3.40373}, {3.05937, 0.0417194, 1.30474, 1.56848,
5374.06}, {0.729741, 0.942849, 0.990381,
0.803124, -1.61932}, {3.86653, 3.23461, 1.45103,
1.27452, -0.888884}, {4.21975, 3.5693, 1.78065,
1.96737, -0.591169}, {3.7998, 2.03796, 1.84843,
0.0281808, -64.849}, {4.78858, 4.16817, 0.255213, 0.861542,
0.0208193}, {2.49005, 0.608272, 0.556096, 0.453426,
12.6624}, {1.69236, 0.85002, 0.114269, 1.89718,
3.24608}, {1.37257, 0.451485, 0.291911, 1.00561, 7.80971}};


I want to know how to find formulas from data.

The answer I expect is - b^3 c + a^2 b d - a b^2 d - 2 b^2 c d + a^2 d^2 - b^3 d e - b^2 d^2 e.

Because I generated the data according to this formula.

I have tried InterpolatingPolynomial, but since f==0, so it will just return 0.

• i am confused as what are the parameters and what are the variables. normally, letters like $x,y,z$ etc.. are used for variables and letters like $a,b,c$ etc... are used for parameters. I see here only $a,b,c$ and no $x,y,z$. Is this meant to be a polynomial in $x$ only? – Nasser Apr 27 '20 at 16:02