# Certification

We provide support for certifying non-singular solutions to polynomial systems. The details of the implementation described in the article

Breiding, P., Rose, K. and Timme, S. "Certifying zeros of polynomial systems using interval arithmetic." arXiv:2011.05000

## Certify

HomotopyContinuation.certifyFunction
certify(F, solutions, [p, certify_cache]; options...)
certify(F, result, [p, certify_cache]; options...)

Attempt to certify that the given approximate solutions correspond to true solutions of the polynomial system $F(x;p)$. The system $F$ has to be an (affine) square polynomial system. Also attemps to certify for each solutions whether it approximates a real solution. The certification is done using interval arithmetic and the Krawczyk method[Moo77]. Returns a CertificationResult which additionall returns the number of distinct solutions. For more details of the implementation see [BRT20].

Options

• show_progress = true: If true shows a progress bar of the certification process.
• max_precision = 256: The maximal accuracy (in bits) that is used in the certification process.
• compile = false: See the solve documentation.

Example

We take the first example from our introduction guide.

@var x y
# define the polynomials
f₁ = (x^4 + y^4 - 1) * (x^2 + y^2 - 2) + x^5 * y
f₂ = x^2+2x*y^2 - 2y^2 - 1/2
F = System([f₁, f₂], variables = [x,y])
result = solve(F)
Result with 18 solutions
========================
• 18 paths tracked
• 18 non-singular solutions (4 real)
• random seed: 0xcaa483cd
• start_system: :polyhedral

We see that we obtain 18 solutions and it seems that 4 solutions are real. However, this is based on heuristics. To be absolute certain we can certify the result

certify(F, result)
CertificationResult
===================
• 18 solution candidates given
• 18 certified solution intervals (4 real, 14 complex)
• 18 distinct certified solution intervals (4 real, 14 complex)

and see that there are indeed 18 solutions and that they are all distinct.

source

## CertificationResult

The result of certify is a CertificationResult:

## SolutionCertificate

A CertificationResult contains in particular all SolutionCertificates:

HomotopyContinuation.is_certifiedFunction
is_certified(certificate::AbstractSolutionCertificate)

Returns true if certificate is a certificate that certified_solution_interval(certificate) contains a unique zero.

source
HomotopyContinuation.is_realMethod
is_real(certificate::AbstractSolutionCertificate)

Returns true if certificate certifies that the certified solution interval contains a true real zero of the system. If false is returned then this does not necessarily mean that the true solution is not real.

source
HomotopyContinuation.is_complexMethod
is_complex(certificate::AbstractSolutionCertificate)

Returns true if certificate certifies that the certified solution interval contains a non-real complex zero of the system.

source
HomotopyContinuation.is_positiveMethod
is_positive(certificate::AbstractSolutionCertificate)

Returns true if is_certified(certificate) is true and the unique zero contained in certified_solution_interval(certificate) is real and positive.

source
HomotopyContinuation.certified_solution_intervalFunction
certified_solution_interval(certificate::AbstractSolutionCertificate)

Returns an Arblib.AcbMatrix representing a vector of complex intervals where a unique zero of the system is contained in. Returns nothing if is_certified(certificate) is false.

source
HomotopyContinuation.certified_solution_interval_after_krawczykFunction
certified_solution_interval_after_krawczyk(certificate::ExtendedSolutionCertificate)

Returns an Arblib.AcbMatrix representing a vector of complex intervals where a unique zero of the system is contained in. This is the result of applying the Krawczyk operator to certified_solution_interval(certificate). Returns nothing if is_certified(certificate) is false.

source
• Moo77Moore, Ramon E. "A test for existence of solutions to nonlinear systems." SIAM Journal on Numerical Analysis 14.4 (1977): 611-615.
• BRT20Breiding, P., Rose, K. and Timme, S. "Certifying zeros of polynomial systems using interval arithmetic." arXiv:2011.05000.