Diophantine equations solver

Note:
1.Enter polynomial only. "=" is not available.
2.Please use + for addition, * for multiplication(this can be omitted, e.g. 2x will be read as 2*x, xy will be read as x*y, etc.) and don't input integer multiple integer such as 3*4.
3.Please use ^ for power, and () for brackets. (...)2x is not available.
4.Variables are either one letter from a to z, or one letter and one digit, e.g. x0, x1, or x9.
5.No mixture of variables such as x and x1 in one polynomial.
6.Variable including 'n','m' are not available because \(n_{0}\),\(n_{1}\),\(n_{2}\),... and \(m_{0}\),\(m_{1}\),\(m_{2}\),... represent the integer parameters in solution.
7.Several polynomials can be entered to be solved simultaneously where they are separated by ",".
8.Equation systems can be solved by entering "sys;" at the beginning of polynomials described in 7.

Method 1: Univariable case
Example: \(x^{5}+2x^{3}+1=0\)

Method 2: Parameter form
Example: \(x+y^{2}+3=0\)

Method 3: Zero factorization
Example: \((x+y)(y+z)=0\)

Method 4: Integer subtraction for factorization
Example: (Used in solving process)

Method 5: None zero factorization
Example: \((x+y)(y+z)=1\)

Method 6: Equation system
Example: \(\left\{ \begin{aligned}&x+2y+2=0\\&x^2+y+1=0\end{aligned} \right.\)

Method 7: No solution verification
Example: \(x^{2}+y^{2}+1=0\)

Method 8: Verification of maximum solution length
Example: \(x^{2}+y^{2}-1=0\)

Method 9: Verification of maximum of minimum of variables in solution
Example: \(xyz+y+z=0\)

Method 10: Linear case
Example: \(8x-3y+5z+22=0\)

Method 11: Rational family case
Example: \(xy+zw+2q=0\)

Method 12: Two monomials case
Example: \(x^2=yz\)

Method 13: Linear substitution
Example: (Used in solving process)

Method 14: Variable division
Example: \(xy+y^{5}z+z^{2}=0\)

Method 15: Vieta jump
Example: \(x^2+xy-y^2+x=0\)

Method 16: Pell's equation
Example: \(x^2-2y^2=1\)

Method 17: General quadratic equation
Example: \(2x^2-y^2+x-1\)

Method 18: Mordell's equation
Example: \(y^2=x^3+3\), \(x^3+y^2+1=0\), \(x^3+2x^2+y^2=0\)

Method 19: Equation with discriminant
Example: \(x^4+x-y^2=0\), \(x^4+x^3-y^2=0\)

Method 20: Optimization with bound
Example: \(y^3=x^3-x\)

Method 21: 2-variables homogeneous case
Example: \(y^3+3x^3-4xy^2\)

Method 22: Elliptic curve substitution
Example: \(x^4+y^6-2\)

Method 23: Thue equation
Example: \(y^3+3x^3-4xy^2+3\)

Method 24: 2-variables linear case
Example: \(x^3+2x^2y-y+1\)

Method 25: 2-variables reducible case
Example: \(y^3+y=2x^3+2\)

Method 26: 2-variables no power terms case
Example: \(xQ(x,y)+ay+b=0\)

Method 27: 2-variables special homogeneous case
Example: \(x^3-2x+y^3=0\), \(x^3-x+y^3+y=0\)

Method 28: Equation reducible over \(\mathbb{C}\)
Example: \(x^3+2y^3-6xy+4=0\)

Method 29: Equations of genus 0
Example: \(4x^3+2y^3+3y^2=0\)

Method 30: Equation satisfying Runge's condition
Example: \(x^2(2x+3y)+y^2+3x^2-9=0\)