# An efficient algorithm to find a linear transformation between two ternary quadratic forms

Let $$\mathbb{F}_p$$ be a prime finite field for $$p > 2$$. Consider two ternary quadratic forms $$Q_1\!: x^2 - a_1(t)y^2 - b_1(t)z^2,\\ Q_2\!: x^2 - a_2(t)y^2 - b_2(t)z^2$$ over the field $$\mathbb{F}_p(t)$$ of rational functions with coefficients from $$\mathbb{F}_p$$. For simplicity let $$a_1, a_2, b_1, b_2 \in \mathbb{F}_p[t]$$ are polynomials without multiple roots and $$a_1$$, $$b_1$$ (respectively $$a_2$$, $$b_2$$) have no common roots.

Is there an efficient algorithm to find a linear transformation (over $$\mathbb{F}_p(t)$$) between $$Q_1$$ and $$Q_2$$ if it exists?

There is the theory that relates such forms and quaternion algebras (see, for example, $$\S$$1.4 in Book of Gille, Szamuely - Central Simple Algebras and Galois Cohomology). For example, for any non-zero polynomial $$f \in \mathbb{F}_p[t]$$ and $$p > 2$$ the following quadratic forms are isomorphic: $$Q_1\!: x^2 - y^2 - f(t)z^2,\\ Q_2\!: x^2 - y^2 - z^2$$ This is true, because $$Q_1$$ can be reduced to the quadratic form $$Q_3\!: x^\prime y^\prime-(z^\prime)^2$$ by the transformation $$x := x^\prime+\frac{y^\prime}{4f},\qquad y := x^\prime-\frac{y^\prime}{4f},\qquad z := \frac{z^\prime}{f}.$$ It is well known that any two conics (including $$Q_2$$, $$Q_3$$) over a finite field are isomorphic.

• What do you mean by a linear transformation? Is it $x'=c_0x+c_1$, $y'=c_2x+c_3$, $z'=c_4z+c_5$? If so it seems easy to prove that there can never be any non-trivial linear transformation, since the coefficients of $x,y,z,1$ are zero. – D.W. Jan 21 at 16:51
• I mean a non-degenerate projective transformation: $x := c_1x + c_2y + c_3z$, $y := d_1x + d_2y + d_3z$, $z := e_1x + e_2y + e_3z$, where coefficients from $\mathbb{F}_p(t)$. – Dima Koshelev Jan 21 at 17:15
• Still seems impossible for the same reasons. Can you edit the question to give an example of two such quadratic forms where a linear transformation does exist? – D.W. Jan 21 at 17:18
• I added the comments. – Dima Koshelev Jan 21 at 17:59
• You claim that they are isomorphic in the question; how can you know that, if you can't find such a transformation? I think you should put some more effort into your question first. I suggest trying an example and try proving whether such a transformation exists. You should be able to write down a system of 8 equations on the 9 unknowns $c_1,c_2,c_3,d_1,d_2,d_3,e_1,e_2,e_3$ and then see if any solution exists, and thus whether any such linear transformation exists. I think you should also edit the question to show your definition of "linear transformation" in the question. – D.W. Jan 21 at 23:08