Nemtsov function

\(y=\mathrm{nem}_q(x)\)

Complex maps: \(u+\mathrm i v=f(x+\mathrm i y)\) for \(f=\mathrm{nem}_0\), \(f=\mathrm{ArqNem}_0\) (top) and for \(f=\mathrm{nem}_2\), \(f=\mathrm{ArqNem}_2\) (bottom)

The Nemtsov function is a special kind of polynomial, suggested as an example of a transfer function in the book «Superfunctions» . The description is also available as the Mizugadro Preprint .

The Nemtsov function \(y=\mathrm{nem}_q(x)= x+x^3+q\,x^4\) is shown in the figure at right for various \(q\ge 0\).

Complex maps of \(\mathrm{nem}_q\) are shown in the left column of the figure below. The right column shows similar maps for the inverse function \(\mathrm{ArqNem}_q=\mathrm{nem}_q^{-1}\).

This article describes the Nemtsov function and its related functions: the inverse function ArqNem, the superfunction SuNem, and the Abel function AuNem.

Motivation

The Nemtsov function serves as an example of a real-holomorphic transfer function with a real fixed point, where the usual regular iteration method for constructing a superfunction cannot be applied directly. The editor did not find any simpler example of this type other than this specific fourth-order polynomial.

The function was introduced as an attempt to construct an “exotic’’ transfer function for which a growing real-holomorphic superfunction could not be produced by the usual methods appearing in the «Table of superfunctions». This attempt failed — the superfunction for the Nemtsov function *can* be constructed, and is described below.

The expansion of the Nemtsov function at its fixed point begins with a linear term whose coefficient is unity. This prevents the use of standard regular iteration , which works for the exponential to base \(\sqrt{2}\).

The situation is similar to the exponential to base \(\exp(1/\mathrm e)\) , but the Nemtsov function lacks the quadratic term, so the exotic iteration of cannot be used as is.

A closer analogy is the case of the sine function , but sine is antisymmetric, \(\sin(-z)=-\sin(z)\), which simplifies its analysis. The Nemtsov function for \(q>0\) has no such symmetry, leading initially to doubts whether a superfunction could be constructed.

Eventually, the construction succeeded.

B. Nemtsov

The need for a special name for this function emerged on 2015-02-27, the day when Putin killed Nemtsov. As of 2025, no other scientific concept attached to that event has appeared. The total corruption in Russia prevents professional investigation of the crime. Thus, the family name “Nemtsov’’ serves as a historical timestamp.

By 2025, no better notation for this polynomial has been proposed. The function symbol is written as lowercase \(\mathrm{nem}\), following the convention for mathematical functions, even though “Nemtsov’’ is a proper name. Capitalization is used only when distinguishing it from derived functions such as ArqNem, AuNem, and SuNem.

Definition and notations

Let \(q\ge 0\) be a real parameter. The Nemtsov function is defined for complex argument \(z\) by

\[ \mathrm{nem}_q(z)=z+z^3+q\,z^4 \tag{1} \]

For \(q>0\), the algorithms described in the first Russian edition of «Суперфункции» (2014) cannot be applied in their original form; a small generalization is required. Thus the function \(\mathrm{nem}_q\) is treated as a transfer function.

This appears to be the last remaining attempt (as of 2025) to produce a real-holomorphic, growing transfer function whose superfunction cannot be constructed by methods already known in the literature.

But again, the attempt fails: both the superfunction SuNem and the Abel function AuNem are constructed using a method similar to that used for the sine function.

Associated functions:

Inverse function: \(\mathrm{ArqNem}_q\), satisfying \[ \mathrm{nem}_q(\mathrm{ArqNem}_q(z)) = z \]

Superfunction \(\mathrm{SuNem}_q\), satisfying \[ \mathrm{SuNem}_q(z+1) = \mathrm{nem}_q(\mathrm{SuNem}_q(z)) \]

Abel function: \(\mathrm{AuNem}_q=\mathrm{SuNem}_q^{-1}\), satisfying

\[ \mathrm{AuNem}_q(\mathrm{nem}_q(z)) = \mathrm{AuNem}_q(z) + 1 \]

Several inverse functions exist: ArcNem, ArkNem, and ArqNem. They differ in branch-cut placement. ArqNem turns out to be the correct one for defining holomorphic non-integer iterates near the positive real axis; therefore it is adopted as the default inverse.

Inverse function

   

            \(x\!+\!\mathrm i y\!=\!\mathrm{NemBra}(q) ~\) and \(~ x\!+\!\mathrm i y\!=\!\mathrm{NemBran}(q)\)

To construct the inverse function in the complex plane, one must locate the saddle points and choose suitable branch cuts. In the complex maps shown previously, the yellow lines indicate the branch cuts for functions \(\mathrm{ArqNem}_0\) and \(\mathrm{ArqNem}_2\). These lines connect the two complex branch points of \(\mathrm{ArqNem}_q\).

For real \(q\), the Nemtsov function is real-holomorphic in the whole complex plane:

\[ \mathrm{nem}_q(z^{*}) = \mathrm{nem}_q(z)^{*}. \]

For positive real arguments, the function grows monotonically. This monotonicity is inherited by its inverse \(\mathrm{ArqNem}_q\), its Abel function \(\mathrm{AuNem}_q\), and its real iterates.

The inverse function \(\mathrm{ArqNem}_q\) has complex branch points. One of them is described explicitly by the function NemBran. At \(z=\mathrm{NemBran}(q)\), the derivative of \(\mathrm{ArqNem}_q(z)\) becomes infinite.

The branch points arise from the complex solutions \(A\) of the equation

\[ \mathrm{nem}_q'(A) = 0. \]

One such solution is denoted \(\mathrm{NemBra}(q)\), and its image

\[ \mathrm{NemBran}(q) = \mathrm{nem}_q(\mathrm{NemBra}(q)). \]

For positive \(q\), both real and imaginary parts of \(\mathrm{NemBra}(q)\) and \(\mathrm{NemBran}(q)\) are small (well below unity).

Once \(\mathrm{NemBra}\) is implemented, an efficient algorithm for inverse functions becomes possible. Three versions—ArcNem, ArkNem, and ArqNem—differ only in branch-cut structure. ArqNem is best suited for constructing holomorphic non-integer iterates near the positive real axis.

The C++ implementation of \(\mathrm{ArqNem}_q\) is available as arqnem.cin. Parameter \(q\) is stored in the global variable `Q`. Before evaluating \(\mathrm{ArqNem}_q(z)\), the corresponding branch point must be computed via nembran.cin and stored in global variables (real and imaginary parts).

Superfunction

\(y=\mathrm{SuNem}_{q}(x)\) for various \(q\)

  \(u+\mathrm i v=\mathrm{SuNem}_0(x+\mathrm i y)\)   \(u+\mathrm i v=\mathrm{SuNem}_2(x+\mathrm i y)\)

For the transfer function \(\mathrm{nem}_q\), the superfunction \(\mathrm{SuNem}_q\) is the real-holomorphic solution \(F\) of the transfer equation

\[ F(z+1) = \mathrm{nem}_q(F(z)), \]

with the asymptotic condition at \(-\infty\):

\[ F(z) = \frac{1}{\sqrt{-2z}} \left( 1 - \frac{q}{\sqrt{-2z}} + O\!\left( \frac{\ln(-z)}{z} \right) \right), \qquad z\to -\infty. \]

To fully specify \(\mathrm{SuNem}_q\), an additional normalization is imposed:

\[ \mathrm{SuNem}_q(0) = 1. \]

The plot \(y=\mathrm{SuNem}_q(x)\) appears at left for several values of \(q\). The function grows monotonically from \(0\) at \(-\infty\), reaches \(1\) at argument \(0\), and then increases rapidly for positive arguments. Larger \(q\) produces faster growth.

Complex maps \(u+\mathrm i v=\mathrm{SuNem}_q(x+\mathrm i y)\) are displayed in figures at right for \(q\!=\!0\) and \(q\!=\!2\). Regions with high density of contours (in vicinity of positive part of the real axis) are left empty; this is convenient mode to show complex maps of functions with fast growth.

To construct \(\mathrm{SuNem}_q\), one first constructs any superfunction \(F\) with the correct asymptotics, and then defines

\[ \mathrm{SuNem}_q(z) = F(x_1 + z), \]

where \(x_1\) is the real number satisfying \(F(x_1)=1\).

For transfer function \(\mathrm{nem}_q\), the \(\mathrm{SuNem}_q\) seems to be the only real-holomorphic superfunction that takes value unity at zero and approaches zero at \(x\pm \mathrm i \infty\); in this sense, \(\mathrm{SuNem}_q\) is the simplest superfunction.

Abel function

\(y=\mathrm{AuNem}_q(x)\) for \(q=0\), \(1\), \(2\)

For the Abel function of the Nemtsov function, the notation AuNem is used.

The explicit plot \(y=\mathrm{AuNem}_q(x)\) versus \(x\) is shown at left for \(q=0\), \(1\), and \(2\). This plot coincides with reflecting the plots of \(\mathrm{SuNem}_q\) across the bisector of the first quadrant. This symmetry provides a numerical test:

\[ \mathrm{SuNem}_q(\mathrm{AuNem}_q(x)) = x, \qquad \mathrm{AuNem}_q = \mathrm{SuNem}_q^{-1}. \]

As the inverse of the superfunction, the Abel function satisfies the Abel equation

\[ \mathrm{AuNem}_q(\mathrm{nem}_q(z)) = \mathrm{AuNem}_q(z) + 1. \]

The normalization is

\[ \mathrm{AuNem}_q(1) = 0, \]

because \(\mathrm{SuNem}_q(0)=1\).

The asymptotic expansion of \(\mathrm{AuNem}_q\) near \(0\) follows from inverting the asymptotics of \(\mathrm{SuNem}_q\) at \(-\infty\).
This expansion gives way to the efficient implementation, in analogy with the regular iteration.

Iterates

  \(y=\mathrm{nem}_0^{\,n}(x)\) for various \(n\)   \(y=\mathrm{nem}_1^{\,n}(x)\) for various \(n\)   \(y=\mathrm{nem}_2^{\,n}(x)\) for various \(n\)

Using the functions \(\mathrm{SuNem}_q\) and \(\mathrm{AuNem}_q\), the \(n\)-th iterate of the Nemtsov function can be expressed as

\[ \mathrm{nem}_q^{\,n}(z) = \mathrm{SuNem}_q\!\big(n + \mathrm{AuNem}_q(z)\big). \]

Here, the number \(n\) of iterate has no need to be integer; the function can be iterated real or even complex number of times.

Plots of the iterates for \(q=0\), \(q=1\), and \(q=2\) appear at right. For positive integer \(n\), the iterate grows rapidly; for negative \(n\), the growth is slow.

The zeroth iterate (\(n=0\)) is the identity function. In the figures, this curve appears as a green line.

Because \(\mathrm{ArqNem}_q\) has a singularity at \(0\), non-integer iterates are not defined at \(0\) or on the negative real axis, though they approach \(0\) from the right.

At least in some vicinity of the positive part of the real axis (moderate values of \(\Im(z)\)), the iterates satisfy the symmetry \[ \mathrm{nem}_q^{-n}(z) = \mathrm{ArqNem}_q^{\,n}(z). \] This relation holds until to reach the first branch cut of function AuNem\(_q\).

Applications

The Nemtsov function was proposed as a candidate transfer function for which the superfunction and the Abel function would be difficult to construct using the exotic iterate at its fixed point \(0\). The difficulty lay mostly in constructing the correct inverse function \(\mathrm{ArqNem}_q\), choosing branch cuts appropriately, and recognizing that \(\mathrm{ArqNem}_q\) rather than \(\mathrm{ArcNem}\) or \(\mathrm{ArkNem}\) should serve as the default inverse.

Other inverse functions differ only in branch-cut placement and do not yield a real-holomorphic Abel function on as wide a domain.

Once the inverse iterates converge to the fixed point, the exotic iteration becomes straightforward. The same approach can be used for other exotic transfer functions whose expansion at the fixed point begins with a linear term and a cubic term (the cubic coefficient can always be normalized to \(1\)). The required transformation appears in the last row of the Table of superfunctions.

Warning

The name “Nemtsov function’’ was chosen as a mnemonic connected to a notable event of the 21st century.

It is not intended as a political statement or as an attempt to appeal to the Russian usurper or his accomplices. The ongoing rise of corruption indicates that moral appeals are meaningless in that context, even while some impostors pretend they can “end war in 24 hours’’ by appeasing aggression and war crimes.

Nevertheless, the editor reserves the right to use the most convenient system of notation.

Acknowledgement

ChatGPT helped to improve this article.

References

Keywords

«Abel function», «ArqNem», «AuNem», «Book», «C++», «Exotic iterate», «Exotic iteration», «Latex», «Mathematica», «Nemtsov function», «Putin killed Nemtsov», «SuNem», «Superfunction», «Superfunctions», «Table of superfunctions»,

«Немцов Борис Ефимович», «Путин убил Немцова»,