I was wondering if there is a program or website that solved a system of nonlinear equations I was trying a system that looks like this $$a+b = cd , a+c = bd ,a+d = bc$$ , ... it's like a pattern that share the idea of For all sets S⊆U such that $|S|=m$ and $|U|=2m $ , $$∏_{i∈S} qi = \sum_{i∈U∖S} qi $$
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3$\begingroup$ Yes, there are many CAS you can use: for emxaple Macaulay2, mathematica, maple, GAP, sage, reduce, singular etc., see this list. Please include your equations. We may be able to solve them immediately. Use variables $x_i$ for simplicity. $\endgroup$– Dietrich BurdeMar 21, 2023 at 15:23
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$\begingroup$ @DietrichBurde sry about my Q not being clear I added the problem I was trying to solve $\endgroup$– hn_garaMar 21, 2023 at 15:32
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$\begingroup$ So the variables are $a,b,c,d,\ldots $, and the solutions is sought in integers? What is the domain for solving $a+b=cd$, etc ? Please use MathJax. Here is a tutorial. You should first try for small $m$. $\endgroup$– Dietrich BurdeMar 21, 2023 at 15:33
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$\begingroup$ @Moo thx for help the problem with wolfram that is solve only 4 equations I was trying to solve 6 equations to find the pattern to the problem I just added $\endgroup$– hn_garaMar 21, 2023 at 15:34
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$\begingroup$ @DietrichBurde yes the solution is integers $\endgroup$– hn_garaMar 21, 2023 at 15:35
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The system of equations $$a+b = cd , a+c = bd ,a+d = bc$$ has the solutions $(a,b,c,d)$ as follows: $$ (-b-1,b,-1,-1),\; (1-d,-1,-1,d),\; (1-c,-1,c,-1),\;(d(d-1),d,d,d). $$ This already shows a pattern.
answered Mar 21, 2023 at 15:49
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$\begingroup$ thanks for helping that helped me to solve this problem but I was wondering if there a tool that solves it for large m or any other constraints so I find the pattern faster if I face a problem similar to this $\endgroup$– hn_garaMar 21, 2023 at 15:56
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$\begingroup$ Yes, you can use a CAS. However, not all systems of polynomial equations can be solved so easily, in particular not over the integers. $\endgroup$ Mar 21, 2023 at 15:58
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