Using Newton-Puiseux Algorithm

[RagonEbker]

Hello,

it is not really clear to me how to work with single line equations. For example I want to calculate the Newton-Puiseux Algorithm of a polynomial, i.e. : x**2 + (2*y + 3)*x + y**2 + 3*y + 2 .

If entered into the program in the corresponding format:
1 x**2 + (2*y + 3)*x + y**2 + 3*y + 2;

Is this something that's even supported by the software?

[janverschelde]

No, it is not directly supported, although some of the algorithms needed in the Newton-Puiseux algorithm are implemented. The phcpy package provides a beginning of an interface.

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________________________________ From: RagonEbker ***@***.***> Sent: Friday, August 4, 2023 6:49 PM To: janverschelde/PHCpack ***@***.***> Cc: Subscribed ***@***.***> Subject: [janverschelde/PHCpack] Using Newton-Puiseux Algorithm (Issue #68) Hello, it is not really clear to me how to work with single line equations. For example I want to calculate the Newton-Puiseux Algorithm of a polynomial, i.e. : x2 + (2*y + 3)*x + y2 + 3*y + 2. If entered into the program in the corresponding format: 1 x**2 + (2*y + 3)*x + y**2 + 3*y + 2; Is this something that's even supported by the software? — Reply to this email directly, view it on GitHub<#68>, or unsubscribe<https://github.com/notifications/unsubscribe-auth/AA53KBKRM544UPLC5GZWYE3XTWDBNANCNFSM6AAAAAA3EZ3LVM>. You are receiving this because you are subscribed to this thread.Message ID: ***@***.***>

[LegionMammal978]

I've also been looking for this. I have a moderately large polynomial system with a single parameter that goes to infinity (or equivalently, one that goes to 0), and I just want to know the leading term of each variable w.r.t. the parameter. If PHCpack and phcpy don't support this, would you know of any other tool that does? In the worst case, I guess I could always just take some samples at different parameter values and manually fit a curve.

[RagonEbker]

Hey @LegionMammal978, could you clarify your comment, what do you need exactly, a newton puiseux type algorithm? Because I am working on one at the moment.

[LegionMammal978]

Hey @LegionMammal978, could you clarify your comment, what do you need exactly, a newton puiseux type algorithm? Because I am working on one at the moment.

That is, given a system of parameterized polynomial equations F_1(t, x_1,x_2,...,x_n) = 0, ..., F_n(t, x_1,x_2,...,x_n) = 0, let x_1(t), x_2(t), ..., x_n(t) denote the positive real solution, as a function of t. Determine the asymptotic values x_i(t) ~ c_i*t^α_i as t → ∞. (Though some further experiments have shown that my instances are somewhat too dense for homotopy-continuation methods to give a result within a reasonable time, so perhaps I'll just resort to local methods after all.)