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I am trying to construct a system of two cubic polynomial equations in two variables (x and y) with exactly 9 real solutions using Maple. However, I am having trouble finding the appropriate coefficients for the polynomials to ensure exactly 9 real solutions. Here is the system $$ a_1 x^3 + a_2 x^2 y + a_3 x y^2 + a_4 x^2 + a_5 xy + a_6 y^2 + a_7 x + a_8 y =0 $$ $$ b_1 x^3 +b_2 y^3 + b_3 x^2 y + b_4 x y^2 + b_5 x^2 + b_6 xy + b_7 y^2 + b_8 x + b_9 y =0 $$

I would appreciate any guidance on how to construct such a system, ensuring exactly 9 real solutions with $x$ non-negative. Any insights or examples would be greatly appreciated!

I have also posted this question on MSE (https://math.stackexchange.com/questions/4951785/constructing-a-system-of-two-cubic-polynomial-equations-with-exactly-9-real-solu).

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    $\begingroup$ Gerald Edgar suggested, but deleted, using $(x-1)(x-2)(x-3) = (y-1)(y-2)(y-3) = 0$. One could more generally use any independent linear combinations of two such polynomials; or more interesting, start from such a system (or translate to the simpler-looking $x^3-x = y^3-y = 0$) and deform a bit so the solutions are real but not obvious. $\endgroup$
    – Noam D. Elkies
    Commented Jul 29 at 2:13
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    $\begingroup$ @NoamD.Elkies: I started with $x^3-x = y^3-y = 0$, then noticed the condition $x$ non-negative. Also, it seems the first one has no $y^3$ term. $\endgroup$
    – Gerald Edgar
    Commented Jul 29 at 2:16
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    $\begingroup$ Crossposted from MSE, where he received enough hints in comments $\endgroup$
    – Will Jagy
    Commented Jul 29 at 2:59
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    $\begingroup$ math.stackexchange.com/questions/4951785/… $\endgroup$
    – Will Jagy
    Commented Jul 29 at 3:00
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    $\begingroup$ In general it is considered good etiquette to wait at least a couple of days after one posts on math.SE, and then mention the original question if one does crosspost. This avoids unnecessary duplicate work. $\endgroup$
    – Jules Lamers
    Commented Jul 29 at 8:01

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