# How to solve 29 coupled quadratic equations?

I have a set of 29 coupled quadratic equations, with 29 unknown variables.

Can anyone offer any advice on how I could go about solving this?

3 days of staring at a wall has so far given me no thoughts on how to do this at all.

EDIT: \begin{align} T_1 &= X_1^2 X_2 X_3 X_4 X_5 X_6 \\ T_2 &= X_2^2 X_1 X_3 X_4 X_5 X_6 \\ T_3 &= X_3^2 X_1 X_2 X_4 X_5 X_6 \\ T_4 &= X_4^2 X_1 X_2 X_3 X_5 X_6 \\ T_5 &= X_5^2 X_1 X_2 X_3 X_4 X_6 \\ T_6 &= X_6^2 X_1 X_2 X_3 X_4 X_5 \\ T_7 &= X_1 X_2 X_3 X_4 X_5 X_6 X_7^2 X_8 X_9 X_{10} (1-X_5) \\ T_8 &= X_1 X_2 X_3 X_4 X_5 X_6 X_7 X_8^2 X_9 X_{10} (1-X_5) \\ T_9 &= X_1 X_2 X_3 X_4 X_5 X_6 X_7 X_8 X_9^2 X_{10} (1-X_5) \\ T_{10} &= X_1 X_2 X_3 X_4 X_5 X_6 X_7 X_8 X_9 X_{10}^2 (1-X_5) \\ T_{11} &= X_1 X_2 X_3 X_4 X_5 X_6 \{ (1-X_6) + (1-X_{10})(1-X_5)X_7 X_8 X_9 X_{10} \} X_{11}^2 X_{12} X_{13} X_{14} X_{15} \\ T_{12} &= X_1 X_2 X_3 X_4 X_5 X_6 \{ (1-X_6) + (1-X_{10})(1-X_5)X_7 X_8 X_9 X_{10} \} X_{11} X_{12}^2 X_{13} X_{14} X_{15} \\ T_{13} &= X_1 X_2 X_3 X_4 X_5 X_6 \{ (1-X_6) + (1-X_{10})(1-X_5)X_7 X_8 X_9 X_{10} \} X_{11} X_{12} X_{13}^2 X_{14} X_{15} \\ T_{14} &= X_1 X_2 X_3 X_4 X_5 X_6 \{ (1-X_6) + (1-X_{10})(1-X_5)X_7 X_8 X_9 X_{10} \} X_{11} X_{12} X_{13} X_{14}^2 X_{15} \\ T_{15} &= X_1 X_2 X_3 X_4 X_5 X_6 \{ (1-X_6) + (1-X_{10})(1-X_5)X_7 X_8 X_9 X_{10} \} X_{11} X_{12} X_{13} X_{14} X_{15}^2 \\ T_{16} &= X_1 X_2 X_3 X_4 X_5 X_6 (1-X_6)(1-X_9)X_7 X_8 X_9 X_{10} \} X_{11} X_{12} X_{13} X_{14} X_{15} X_{16}^2 X_{17} \\ T_{17} &= X_1 X_2 X_3 X_4 X_5 X_6 (1-X_6)(1-X_9)X_7 X_8 X_9 X_{10} \} X_{11} X_{12} X_{13} X_{14} X_{15} X_{16} X_{17}^2 \\ T_{18} &= X_1 X_2 X_3 X_4 X_5 X_6 X_7 X_8 X_9 X_{10} \} X_{11} X_{12} X_{13} X_{14} X_{15} X_{16} X_{17} (1-X_9)(1-X_5)(1-X_{16}) X_{18}^2 X_{19} X_{20} X_{21} \\ T_{19} &= X_1 X_2 X_3 X_4 X_5 X_6 X_7 X_8 X_9 X_{10} \} X_{11} X_{12} X_{13} X_{14} X_{15} X_{16} X_{17} (1-X_9)(1-X_5)(1-X_{16}) X_{18} X_{19}^2 X_{20} X_{21} \\ T_{20} &= X_1 X_2 X_3 X_4 X_5 X_6 X_7 X_8 X_9 X_{10} \} X_{11} X_{12} X_{13} X_{14} X_{15} X_{16} X_{17} (1-X_9)(1-X_5)(1-X_{16}) X_{18} X_{19} X_{20}^2 X_{21} \\ T_{21} &= X_1 X_2 X_3 X_4 X_5 X_6 X_7 X_8 X_9 X_{10} \} X_{11} X_{12} X_{13} X_{14} X_{15} X_{16} X_{17} (1-X_9)(1-X_5)(1-X_{16}) X_{18} X_{19} X_{20} X_{21}^2 \\ T_{22} &= X_1 X_2 X_3 X_4 X_5 X_6 \{ (1-X_5) X_7 X_8 X_9 X_{10} [(1-X_{17} + (1-X_9)(1-X_7)X_{16}X_{17}] + (1-X_2)\} X_{22}^2 X_{23} \\ T_{23} &= X_1 X_2 X_3 X_4 X_5 X_6 \{ (1-X_5) X_7 X_8 X_9 X_{10} [(1-X_{17} + (1-X_9)(1-X_7)X_{16}X_{17}] + (1-X_2)\} X_{22} X_{23}^2 \\ T_{24} &= X_1 X_2 X_3 X_4 X_5 X_6 \{ (1-X_3) + (1-X_5)(1-X_8)X_7 X_8 X_9 X_{10} \} X_{24}^2 X_{25} \\ T_{25} &= X_1 X_2 X_3 X_4 X_5 X_6 \{ (1-X_3) + (1-X_5)(1-X_8)X_7 X_8 X_9 X_{10} \} X_{24} X_{25}^2 \\ T_{26} &= X_1 X_2 X_3 X_4 X_5 X_6 \{ (1-X_4) + \{ (1-X_6) + (1-X_{10})(1-X_5))X_7 X_8 X_9 X_{10} X_{11}X_{12} X_{13} X_{14} X_{15} \}(1-X_{12}) + X_7 X_8 X_9 X_{10} X_{11}X_{12} X_{13} X_{14} X_{15}X_{16}X_{17}X_{18}X_{19}X_{20}X_{21}(1-X_{20})(1-X_5)(1-X_9)(1-x_16) +(1-X_25)\{(1-X_3) + (1-X_5)X_7 X_8 X_9 X_{10} (1-X_8) \}X_{24}X_{25} \} X_{26} \\ T_{27} &= X_1 X_2 X_3 X_4 X_5 X_6 \{ (1-X_4) + \{ (1-X_6) + (1-X_{10})(1-X_5))X_7 X_8 X_9 X_{10} X_{11}X_{12} X_{13} X_{14} X_{15} \}(1-X_{12}) + X_7 X_8 X_9 X_{10} X_{11}X_{12} X_{13} X_{14} X_{15}X_{16}X_{17}X_{18}X_{19}X_{20}X_{21}(1-X_{20})(1-X_5)(1-X_9)(1-x_16) +(1-X_25)\{(1-X_3) + (1-X_5)X_7 X_8 X_9 X_{10} (1-X_8) \}X_{24}X_{25} \} X_{27} \\ T_{28} &= \{ X_1 X_2 X_3 X_4 X_5 X_6 \{ (1-X_4) + \{ (1-X_6) + (1-X_{10})(1-X_5))X_7 X_8 X_9 X_{10} X_{11}X_{12} X_{13} X_{14} X_{15} \}(1-X_{12}) + X_7 X_8 X_9 X_{10} X_{11}X_{12} X_{13} X_{14} X_{15}X_{16}X_{17}X_{18}X_{19}X_{20}X_{21}(1-X_{20})(1-X_5)(1-X_9)(1-x_16)+(1-X_25)\{(1-X_3) + (1-X_5)X_7 X_8 X_9 X_{10} (1-X_8) \}X_{24}X_{25} \} \}(1-X_{27}+ X_1 X_2 X_3 X_4 X_5 X_6 X_7 X_8 X_9 X_{10} \} X_{11} X_{12} X_{13} X_{14} X_{15} X_{16} X_{17} (1-X_9)(1-X_5)(1-X_{16}) X_{18} X_{19} X_{20} X_{21}(1-X_20) +(1-X_{13})X_{28}X_{29} X_1 X_2 X_3 X_4 X_5 X_6 \{ (1-X_6) + (1-X_{10})(1-X_5)X_7 X_8 X_9 X_{10} \} X_{11} X_{12} X_{13} X_{14} X_{15} \} X_{28} \\ T_{29} &= \{ X_1 X_2 X_3 X_4 X_5 X_6 \{ (1-X_4) + \{ (1-X_6) + (1-X_{10})(1-X_5))X_7 X_8 X_9 X_{10} X_{11}X_{12} X_{13} X_{14} X_{15} \}(1-X_{12}) + X_7 X_8 X_9 X_{10} X_{11}X_{12} X_{13} X_{14} X_{15}X_{16}X_{17}X_{18}X_{19}X_{20}X_{21}(1-X_{20})(1-X_5)(1-X_9)(1-x_16)+(1-X_25)\{(1-X_3) + (1-X_5)X_7 X_8 X_9 X_{10} (1-X_8) \}X_{24}X_{25} \} \}(1-X_{27}+ X_1 X_2 X_3 X_4 X_5 X_6 X_7 X_8 X_9 X_{10} \} X_{11} X_{12} X_{13} X_{14} X_{15} X_{16} X_{17} (1-X_9)(1-X_5)(1-X_{16}) X_{18} X_{19} X_{20} X_{21}(1-X_20) +(1-X_{13})X_{28}X_{29} X_1 X_2 X_3 X_4 X_5 X_6 \{ (1-X_6) + (1-X_{10})(1-X_5)X_7 X_8 X_9 X_{10} \} X_{11} X_{12} X_{13} X_{14} X_{15} \} X_{29} \end{align}

Here are the equations, my unknowns are the $$X$$ terms. The $$T$$ terms are known.

Both $$T$$ and $$X$$ terms are real and positive.

Thanks for the comments, still attempting to understand what a “Groebner basis“ is so far… EDIT END

• @Taladris Coupled means that in each equation one may have several of the variables present.
– Pp..
Jan 22, 2015 at 18:25
• The key words to read about the topic are: Groebner basis, regular chain, elimination of variables. Alternatively, or also, approximation methods. You can star from here
– Pp..
Jan 22, 2015 at 18:31
• That example is not what is normally called quadratic. A property that does stand out is the small number of monomials.
– Pp..
Jan 22, 2015 at 18:33
• Dare I ask what physical system?
– John
Jan 22, 2015 at 19:27
• The X terms are actually exponentials of the form $e^{-t/\tau}$, which are supposed to represent lifetime decays of excited electron states in a crystal structure. There are 29 state transitions, where the T values are transition rates which have been measured. Apparently measuring things is much easier than modelling them.
– user209848
Jan 22, 2015 at 20:31

In the following, I assume that the $(X_i)$ are real numbers.

Following the Pp.. comment about Grobner basis, we can solve the system constituted with the first $6$ equations when the $(T_i)_{i\leq 6}$ are generically chosen.

We obtain a result in the form $a{X_1}^7+b=0$ and, for every $i\leq 6$, $X_i=c_iX_1$. If the $(T_i)$ and $(X_i)$ are real, then we obtain a sole (and explicit) solution for $(X_i)_{i\leq 6}$.

EDIT 1. Since the $(X_i)_{i\leq 6}$ are known, solve the following blocks of equations (we obtain exactly the same type of equations as above)

i) $\{7,\cdots,10\}$, $a{X_7}^5+b=0$, one solution in $X_7,\cdots,X_{10}$.

ii) $\{11\cdots 15\}$, $aX_{11}^6+b=0$, $0$ or $2$ solutions in $X_{11},\cdots,X_{15}$..

iii) $\{16,17\}$, $aX_{16}^3+b=0$, one solution.

iv) $\{18,\cdots,21\}$, after $\{22,23\}$, after $\{24,25\}$. Equations $26,27$ have degree $1$ in $X_{26},X_{27}$. Each of these equations admits generically a unique solution.

v) Equations $28,29$ are in the form $aX_{28}X_{29}+bX_{28}=c,dX_{28}X_{29}+eX_{29}=f$, $0$ or $2$ solutions.

EDIT 2. Grobner basis theory is useless. Indeed the solution(s) of a system of $n$ equations in the form of our first $6$ equations is: ${X_1}^{n+1}=\dfrac{{T_1}^n}{T_2\cdots T_n}$ and, for every $i\leq n$, $X_i=\dfrac{T_i}{T_1}X_1$.

EDIT 3. (answer to Respawned Fluff). The set of solutions of our system is zero-dimensional over $\mathbb{C}$. Then it is easy to keep only the real solutions. Here the system is block-triangular ; moreover each block (using the solutions of the previous blocks) admits an "effective solution" in the following sense: there is $i$ and a polynomial $P$ of degree $d$ s.t. $P(x_i)=0$ and, for every $j\not= i$, there are polynomials $P_j$ of degre $<d$ s.t. $x_j=P_j(x_i)$. Finally, our system has generically $0$ or $4$ real solutions. This is a simple system and clearly it can be easily solved by the standard softwares under the condition that the chosen order for the unknown is essentially $X_1,\cdots,X_{29}$ ; otherwise, the calculation time is likely to be very long.

At least, 2 teams of researchers are working about this subject: the LIP 6 laboratory (J.C. Faugère) and a group around M. Moreno Maza. The first one studies the so-called semi-regular systems over $\mathbb{C}$ and the second one studies the so-called regular semi-algebraic systems over $\mathbb{R}$.

• Sorry, I simply forgot to write $T_{29}$. It's tagged matrices because I was initially trying to write them out as a matrix and then see if that would make it clearer to me on how to solve them. Also I have to have at least one tag to post the question, no currently existing tags that I could see are applicable to the question ,and I do not have enough whatever points to create a new tag....
– user209848
Jan 22, 2015 at 20:26
• Does this triangularization eventually result in a regular semi-algebraic system? If so the authors of that (fairly recent) paper would probably be happy to hear their method applies to some concrete physics problem that's hard to solve otherwise... Jan 23, 2015 at 10:27