Is this solvable in general.

[jwaltos]

Soliciting advice on how to solve this equation:

-2*x^3+(19/2-60*a-60*b*y)*x^2+(21/2+314*a+30*d*y+30*c+314*b*y)*x+2+205*a+11*c+900*b*y*c+420*a*b*y+210*a^2+210*b^2*y^2+11*d*y+900*a*d*y+900*b*y^2*d+205*b*y+900*a*c


Assuming a, b, c and d are known, what is the best method to resolve the remaining variables: x, y, such that the resulting number is a specific integer.

[xilman]

Originally Posted by jwaltos

Soliciting advice on how to solve this equation:

-2*x^3+(19/2-60*a-60*b*y)*x^2+(21/2+314*a+30*d*y+30*c+314*b*y)*x+2+205*a+11*c+900*b*y*c+420*a*b*y+210*a^2+210*b^2*y^2+11*d*y+900*a*d*y+900*b*y^2*d+205*b*y+900*a*c


Assuming a, b, c and d are known, what is the best method to resolve the remaining variables: x, y, such that the resulting number is a specific integer.

Where does this come from? It's far too complicated for it to be at all likely that you made it up on the spur of the moment so it likely arises in the course of some other algebraic manipulations. Knowing its origin could well make a solution easier to find.

BTW, it's not an equation (I don't see a '=' anywhere within it) but a formula. Someone has to do RDS's job for him these days ...

[retina]

Originally Posted by jwaltos

-2*x^3+(19/2-60*a-60*b*y)*x^2+(21/2+314*a+30*d*y+30*c+314*b*y)*x+2+205*a+11*c+900*b*y*c+420*a*b*y+210*a^2+210*b^2*y^2+11*d*y+900*a*d*y+900*b*y^2*d+205*b*y+900*a*c


Assuming a, b, c and d are known, what is the best method to resolve the remaining variables: x, y, such that the resulting number is a specific integer.

You can replace x and y with any value you like. Simplifying and coalescing all the constants gives this formula:

f(x,y) = a*x^3 + (b+c*y)*x^2 + (d+e*y)*x + f*y^2 + g*y + h

[S485122]

You both forgot the last part of the question : "such that the resulting number is a specific integer."

Rephrasing a bit

Assuming a, b, c and d are known, what is the best method to resolve the remaining variables: x, y in order that the formula yields a specific integer A when
A=-2*x^3+(19/2-60*a-60*b*y)*x^2+(21/2+314*a+30*d*y+30*c+314*b*y)*x+2+205*a+11*c+900*b*y*c+420*a*b*y+210*a^2+210*b^2*y^2+11*d*y+900*a*d*y+900*b*y^2*d+205*b*y+900*a*c

Jacob

[retina]

It looks not to dissimilar from an elliptic curve.

[jwaltos]

Thank you for the responses.
I developed this equation as part of an integer factorization toolkit. Complex and rational values apply to `a,b,c,d,x,y` which can also resolve this equation to an integer value.

I can solve for the `a,b,c,d` values in a deterministic manner but resolving `x,y` for large values of `F` in polynomial time still evades me. I am somewhat stymied and vexed regarding how to solve this equation without using sieving or random processes for large values. Transformations into polar coordinates, complex analysis and differential geometry are valid solution paths. A suitable selection for the values `a,b,c,d,x` will algebraically factor the equivalence into (m*y+c1)*(n*y+c2). ie. a=0:b=1:c=35:d=40:x=1 -->(17*y+15)*(2130*y+97)

F=-2*x^3+(19/2-60*a-60*b*y)*x^2+(21/2+314*a+30*d*y+30*c+314*b*y)*x+2+205*a+11*c+900*b*y*c+420*a*b*y+210*a^2+210*b^2*y^2+11*d*y+900*a*d*y+900*b*y^2*d+205*b*y+900*a*c
where F can be any integer.

Basically, my question is, what are the least number of variables within this equation that must be known and what are their numeric limits before it cannot be solved in polynomial time. Conversely, what must be known before this equation can be solved in polynomial time and what tools are required. ie LLL, infinite precision, etc...

[Dr Sardonicus]

Originally Posted by jwaltos

Soliciting advice on how to solve this equation:

-2*x^3+(19/2-60*a-60*b*y)*x^2+(21/2+314*a+30*d*y+30*c+314*b*y)*x+2+205*a+11*c+900*b*y*c+420*a*b*y+210*a^2+210*b^2*y^2+11*d*y+900*a*d*y+900*b*y^2*d+205*b*y+900*a*c


Assuming a, b, c and d are known, what is the best method to resolve the remaining variables: x, y, such that the resulting number is a specific integer.

I second xilman's question, where did this function come from?

I'll call it F(a,b,c,d,x,y)

Knowing what kind of critter this is, would be of considerable help in addressing the question of solving

F(a,b,c,d,x,y) = N

[CRGreathouse]

Originally Posted by jwaltos

Basically, my question is, what are the least number of variables within this equation that must be known and what are their numeric limits before it cannot be solved in polynomial time.

You're asking about a bivariate Diophantine cubic equation. But even bivariate Diophantine quadratics generally take more than polynomial time, so I see no reason to be that optimistic.

[jwaltos]

Originally Posted by xilman

Where does this come from? It's far too complicated for it to be at all likely that you made it up on the spur of the moment so it likely arises in the course of some other algebraic manipulations. Knowing its origin could well make a solution easier to find.

BTW, it's not an equation (I don't see a '=' anywhere within it) but a formula. Someone has to do RDS's job for him these days ...

This equation was derived in my attempts to develop a simple method of factoring integers that can be done in polynomial time without the use of quantum based systems OR to determine that the IFP cannot be resolved in poly time. Some of the literature I have cited in prior posts as well as member posts have contributed to the origin of that expression. It's a simple representative result.
By relaxing the condition in Matiyasevich's theorem for `integer only` solutions, Le Chatelier's principle could be invoked (by analogy) where poly time solutions can be made explicit.

And yes, like a blind squirrel searching for nuts, optimism does help but having a `nose` for certain things prevents that squirrel from starving.
I can't elaborate more without redundancy so I'll just say thanks to those who submitted their input and keep beavering away at this.

[CRGreathouse]

Originally Posted by jwaltos

This equation was derived in my attempts to develop a simple method of factoring integers that can be done in polynomial time without the use of quantum based systems OR to determine that the IFP cannot be resolved in poly time. Some of the literature I have cited in prior posts as well as member posts have contributed to the origin of that expression. It's a simple representative result.
By relaxing the condition in Matiyasevich's theorem for `integer only` solutions, Le Chatelier's principle could be invoked (by analogy) where poly time solutions can be made explicit.

I think you've reduced factoring to a problem harder than factoring.

[R. Gerbicz]

Originally Posted by jwaltos

This equation was derived in my attempts to develop a simple method of factoring integers that can be done in polynomial time without the use of quantum based systems OR to determine that the IFP cannot be resolved in poly time. Some of the literature I have cited in prior posts as well as member posts have contributed to the origin of that expression. It's a simple representative result.
By relaxing the condition in Matiyasevich's theorem for `integer only` solutions, Le Chatelier's principle could be invoked (by analogy) where poly time solutions can be made explicit.

And yes, like a blind squirrel searching for nuts, optimism does help but having a `nose` for certain things prevents that squirrel from starving.
I can't elaborate more without redundancy so I'll just say thanks to those who submitted their input and keep beavering away at this.

I'd say you are overcomplicating this. We can reformulate the factorization problem with a non-linear integer optimization problem: for a given n>1 integer write

min x+y
subject to
x*y=n
x>=0
y>=0
x,y is integer

then n is prime iff the opt=n+1, otherwise n is composite and we'll know a non-trivial factor of n. With this if the above problem is solvable in polynomial time then integer factorization problem is also in poly. (just call it also recursively for the d,n/d factor where d=x).

Using this in some case any solution will give a non trivial factorization, say for
(5*x+2)*(5*y+2)=n. (it'll give a solution if n=4 mod 5 and n has a d=2 mod 5 divisor). Why would be your longer and higher/ degree polynom is easier than mine?