Degree of Jacobian of homogeneous polynomials

What is the degree of the Jacobian (as a polynomial) of 3 homogeneous polynomials in 3 variables of degrees say $m_1, m_2$ and $m_3$ ?

I don't know how to prove that it is independent. In my case the degrees are $2,6$ and $10$. I have tried with some example and I found that the degree is $15$ in this case.

• One idea would be to test the proposition that the answer (degree of the Jacobian) does not depend on the particular polynomials, but can be found from the limited information given, by checking a few cases. Have you tried this? – hardmath Jul 7 '16 at 22:52
• I don't know how to prove that it is independent. In my case the degrees are $2,6$ and $10$. I have tried with some example and I found that the degree is 15 in this case. – Karthik Jul 8 '16 at 9:20
• See my Answer, and note that $2+6+10-3 = 15$. – hardmath Jul 8 '16 at 11:43
• where is your answer ? – Karthik Jul 8 '16 at 16:53
• I realized I'd omitted treating the important case that the Jacobian might be identically zero, so I temporarily deleted the Answer while adding that material. It is back now. – hardmath Jul 8 '16 at 16:57

Having done an example or two should give you a feeling for what can be said about the degree of the Jacobian knowing only the homogeneous degrees of the three polynomials are $m_1,m_2,m_3$ in three variables (say) $x,y,z$.

The Jacobian is the determinant of the $3\times 3$ matrix of first partial derivatives of the three polynomials.

The first partial derivatives of a polynomial of homogeneous degree $m$ will be polynomials of homogeneous degree $m-1$. So in the first row of the $3\times 3$ matrix the partials have degree $m_1-1$, in the second row $m_2-1$, and in the third row $m_3-1$. The product of three polynomials of homogeneous degrees $m_1-1,m_2-1,m_3-1$ will be of homogeneous degree (their sum of degrees) $m_1+m_2+m_3-3$.

Collecting like terms, any signed sum of polynomials of homogeneous degree $m_1+m_2+m_3-3$ will again be of that form, unless it happens to be identically zero.

An example of this with three polynomials of homogeneous degrees $2,6,10$ resp. is $p_1 = x^2, p_2 = y^3 z^3, p_3 = x^4 y^3 z^3$. Their Jacobian is:

$$\mathbf{J} = \det \begin{bmatrix} 2x & 0 & 0 \\ 0 & 3y^2z^3 & 3y^3z^2 \\ 4x^3y^3z^3 & 3x^4y^2z^3 & 3x^4y^3z^3 \end{bmatrix}$$

A brief if tedious calculation (expanding by minors) shows that $\mathbf{J}\equiv 0$. This can be inferred from the functional relationship:

$$p_3 = p_1^2 p_2$$

which implies the polynomial system is not invertible (uniquely solvable for $x,y,z$) at any point in $\mathbb{R}^3$, and therefore vanishes at every point.

However if the Jacobian computed as above is nonzero, it will be homogeneous of degree $m_1+m_2+m_3-3$ because there exist nonzero terms of that degree.