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Michael Schindler
Michael Schindler
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Michael Schindler
Michael Schindler
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In the context of physics, I stumbled over the following problem: I have $N$ equations, all are quadratic in a single scalar, real variable $x$: \begin{eqnarray} 0 &= A_1x^2 + B_1x + C_1 \\ &\dots\\ 0 &= A_Nx^2 + B_Nx + C_N \end{eqnarray} Also the coefficients $A_n,B_n,C_n$ are real scalars. For a solution $x$ to exist, it is necessary that the coefficients $A_n,B_n,C_n$ satisfy compatibility conditions. For small $N$ I can obtain them numerically from Gröbner bases by eliminating $x$. For example, for $N=2$, I obtain \begin{equation} 0 = C_1C_1 A_2A_2 + A_1A_1 C_2C_2 + B_1B_1 C_2A_2 + C_1A_1 B_2B_2 - C_1B_1 A_2B_2 - C_1A_1 C_2A_2 - C_1A_1 A_2C_2 - B_1A_1 B_2C_2 \end{equation} For higher $N$ there are similar other quadraticquartic compatibility equations, but I obtain also cubic ones and others of order five. The number of these conditions grows rapidly with increasing $N$, and I would like to understand their structure.

I would be grateful if someone could help me to understand the structure of these compatibility conditions, or point me to the relevant literature. I did not find a publication on this, and I cannot estimate whether this is standard textbook stuff or not known at all.

In the context of physics, I stumbled over the following problem: I have $N$ equations, all are quadratic in a single scalar, real variable $x$: \begin{eqnarray} 0 &= A_1x^2 + B_1x + C_1 \\ &\dots\\ 0 &= A_Nx^2 + B_Nx + C_N \end{eqnarray} Also the coefficients $A_n,B_n,C_n$ are real scalars. For a solution $x$ to exist, it is necessary that the coefficients $A_n,B_n,C_n$ satisfy compatibility conditions. For small $N$ I can obtain them numerically from Gröbner bases by eliminating $x$. For example, for $N=2$, I obtain \begin{equation} 0 = C_1C_1 A_2A_2 + A_1A_1 C_2C_2 + B_1B_1 C_2A_2 + C_1A_1 B_2B_2 - C_1B_1 A_2B_2 - C_1A_1 C_2A_2 - C_1A_1 A_2C_2 - B_1A_1 B_2C_2 \end{equation} For higher $N$ there are similar other quadratic compatibility equations, but I obtain also cubic ones and others of order five. The number of these conditions grows rapidly with increasing $N$, and I would like to understand their structure.

I would be grateful if someone could help me to understand the structure of these compatibility conditions, or point me to the relevant literature. I did not find a publication on this, and I cannot estimate whether this is standard textbook stuff or not known at all.

In the context of physics, I stumbled over the following problem: I have $N$ equations, all are quadratic in a single scalar, real variable $x$: \begin{eqnarray} 0 &= A_1x^2 + B_1x + C_1 \\ &\dots\\ 0 &= A_Nx^2 + B_Nx + C_N \end{eqnarray} Also the coefficients $A_n,B_n,C_n$ are real scalars. For a solution $x$ to exist, it is necessary that the coefficients $A_n,B_n,C_n$ satisfy compatibility conditions. For small $N$ I can obtain them numerically from Gröbner bases by eliminating $x$. For example, for $N=2$, I obtain \begin{equation} 0 = C_1C_1 A_2A_2 + A_1A_1 C_2C_2 + B_1B_1 C_2A_2 + C_1A_1 B_2B_2 - C_1B_1 A_2B_2 - C_1A_1 C_2A_2 - C_1A_1 A_2C_2 - B_1A_1 B_2C_2 \end{equation} For higher $N$ there are similar other quartic compatibility equations, but I obtain also cubic ones and others of order five. The number of these conditions grows rapidly with increasing $N$, and I would like to understand their structure.

I would be grateful if someone could help me to understand the structure of these compatibility conditions, or point me to the relevant literature. I did not find a publication on this, and I cannot estimate whether this is standard textbook stuff or not known at all.

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Michael Schindler
Michael Schindler
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  • 2

Compatibility conditions for quadratic equations

In the context of physics, I stumbled over the following problem: I have $N$ equations, all are quadratic in a single scalar, real variable $x$: \begin{eqnarray} 0 &= A_1x^2 + B_1x + C_1 \\ &\dots\\ 0 &= A_Nx^2 + B_Nx + C_N \end{eqnarray} Also the coefficients $A_n,B_n,C_n$ are real scalars. For a solution $x$ to exist, it is necessary that the coefficients $A_n,B_n,C_n$ satisfy compatibility conditions. For small $N$ I can obtain them numerically from Gröbner bases by eliminating $x$. For example, for $N=2$, I obtain \begin{equation} 0 = C_1C_1 A_2A_2 + A_1A_1 C_2C_2 + B_1B_1 C_2A_2 + C_1A_1 B_2B_2 - C_1B_1 A_2B_2 - C_1A_1 C_2A_2 - C_1A_1 A_2C_2 - B_1A_1 B_2C_2 \end{equation} For higher $N$ there are similar other quadratic compatibility equations, but I obtain also cubic ones and others of order five. The number of these conditions grows rapidly with increasing $N$, and I would like to understand their structure.

I would be grateful if someone could help me to understand the structure of these compatibility conditions, or point me to the relevant literature. I did not find a publication on this, and I cannot estimate whether this is standard textbook stuff or not known at all.