# Mr. Bayes and the classics: a suggested interpretation

Marcio A. Diniz Department of Statistics, Federal University of São Carlos,
Rod. Washington Luis, km 235, São Carlos, Brazil. Email:
<EMAIL_ADDRESS>Corresponding author. David R. Bellhouse Department of
Statistical and Actuarial Sciences, University of Western Ontario, 1151
Richmond Street, London, Canada. Email<EMAIL_ADDRESS>
###### Abstract

The main hypothesis about Thomas Bayes’s intentions to write his famous Essay
on probability is that he wanted to refute the arguments of David Hume against
the reliability of the occurrence of miracles, published in 1748. In this
paper we argue that it was not Bayes’s intention to rebut Hume but that his
interest on the “inverse problem” came about as result of his study of the
second edition of Abraham De Moivre’s book, The Doctrine of Chances, published
in 1738.

A possible communication of Bayes’s breakthrough might have annoyed De Moivre,
leading to a response written for Bayes in the third edition of De Moivre’s
book, published in 1756. Among other points, the response claims that De
Moivre was the first to solve the mentioned inverse problem. Under this
perspective Richard Price’s letter, written as preface to Bayes’s essay, has a
new interpretation, appearing also as a defense of Bayes premiership on a
satisfactory or proper solution.

keywords: law of large numbers, inverse probability, De Moivre, Thomas Bayes,
Bayesian inference

##  1 Introduction

The motivations that led Thomas Bayes (c. $1701-1761$) to work on his now
famous essay on probability (Bayes, 1763a ) have been searched by historians
for at least the last four decades.111See Stigler, (1983); Dale, (1988);
Edwards, (1986). One hypothesis is that Bayes, a Nonconformist minister,
wanted to refute the arguments of the sceptic philosopher David Hume
($1711-1776$) against the reliability of the occurrence of miracles, published
in 1748 (Hume (1748)). This hypothesis finds support in the prefatory letter
to Bayes’s essay, written by his friend Richard Price ($1723-1791$), also a
Nonconformist minister. In fact, Price used the results proved by Bayes as an
argument to reply to Hume (Price, (1767)), later engaging in correspondence
with the philosopher.

Our goal is to suggest a different perspective on the same story. We will
argue that it was not Bayes’s intention to rebut Hume but that his interest on
the “inverse problem” came about as result of his study of the second edition
of Abraham De Moivre’s book, The Doctrine of Chances (1738). A possible
communication, without details, of Bayes’s breakthrough might have annoyed De
Moivre since this would not be the first time his work was expanded or
corrected by Bayes. This annoyance may have led to a response written for
Bayes in the third edition of De Moivre’s book, published in 1756. We advocate
that this response was not written by De Moivre himself but by the editor of
the third edition of his book, Patrick Murdoch (d. 1774). Murdoch’s response
may have been motivated, or asked for, by De Moivre. Among other points, the
response claims that De Moivre was the first to solve the mentioned inverse
problem. Under this perspective Price’s prefatory letter has a new
interpretation, appearing also as a defense of Bayes premiership on a
satisfactory or proper solution.

The following section reviews the literature to explain what was the “inverse
problem” and how the scholars approached it before Bayes. This will lead us to
a possible solution mentioned by David Hartley in a book published in 1749,
without naming who suggested it. We review the evidence pointing to Bayes as
the one responsible for this anonymous reference. Section 3 brings the
arguments indicating that it was not Bayes’s intention to refute Hume and in
Section 4 we describe how Bayes became interested in probability, eventually
approaching the inverse problem sometime between 1747 and 1750. Section 5
presents the evidence suggesting that Murdoch was the author of the text added
in De Moivre (1756) to, possibly, address Bayes. We conclude by reviewing
Price’s prefatory letter to Bayes’s essay.

##  2 Context: Bernoulli, De Moivre and Hartley

During the seventeenth and first half of the eighteenth centuries, the typical
approach to a probability problem was to assume that the probabilities of some
simple events are known. From these basic assumptions, probabilities of more
complex events were calculated. By 1720, such calculations were done by Jakob
Bernoulli ($1655-1705$), Pierre Rémond de Montmort ($1678-1719$) and De Moivre
($1667-1754$), among others. Later scholars named these methods “direct.” The
great leap forward in the mid eighteenth century would be what David Hartley
($1705-1757$) called the “inverse Problem.” In the context of eighteenth-
century probability calculations this can be stated as: Given the observed
relative frequency of some event in a finite number of similar trials, what
can be said about the underlying probability of this event?

Sometime before 1690, Bernoulli proved the first limit theorem of probability
theory. His result—a particular case of what we now call weak Law of Large
Numbers—was posthumously published in Ars Conjectandi (1713). Bernoulli’s case
of the law of large numbers was also a result in direct probabilities. It
shows how to compute the probability that the relative frequency of repeated
trials, all with a given probability of “success”, will deviate from this
underlying probability when the number of trials increases.222See the Appendix
for the formal statement of Bernoulli’s result and its inverse use. Bernoulli
realized that the result could be applied to situations where events were no
longer considered equiprobable, opening a wide range of applications to
probability theory outside the scope of games of chance.

Daston, (1988) argues that passages from Part IV of Bernoulli, (1713) and,
more explicitly, Jakob’s correspondence with Gottfried Leibniz ($1646-1716$),
indicate that Bernoulli viewed the inverse use of his theorem as a way to
solve333A formal solution of the inverse problem, as stated above, requires
the computation of a conditional probability. See the Appendix for the
differences between Bernoulli’s law, its inverse and Bayes’s theorem. the
inverse problem. Leibniz thought the solution was not satisfactory and raised
some points against it. Bernoulli’s rebuttals to Leibniz were eventually
published in the Ars Conjectandi.444See Bernoulli,, 1713, p. 227 and
Bernoulli,, 2006, p. 329-30. Hacking, (1971, 1975) and Dale, (1988) say that
even though Bernoulli’s intentions can be inferred from his book and letters,
he did not claim to have proved the explicit result. In any case, as Dale,
(1988) comments, at Jakob Bernoulli’s death “one was left with a careful proof
of the direct theorem and a hint at the inverse result.”

De Moivre arrived at his normal approximation to the binomial due to the
answer he gave to a challenge problem proposed to him by Alexander Cuming in
1721. Comments from De Moivre, (1738) on this approximation show that he also
touched on the inverse uses of Benoulli’s limit theorem.

De Moivre’s work on the subject was firstly discussed in Book V of the
Miscellanea Analytica, De Moivre, (1730), where he transcribed several
paragraphs from Bernoulli, (1713). To his conclusions on the normal
approximation to binomial probabilities, he added further work refining
Bernoulli’s computations in a pamphlet called Approximatio ad Summam
Terminorum Binomii $\overline{a+b}|^{n}$ in Seriem expansi, published at his
own expense in 1733 and distributed to a few friends.

The second edition of The Doctrine of Chances, published in 1738, brings the
English translation of the Approximatio exactly after the problems proposed by
Cuming in 1721.555See De Moivre,, 1738, p. 231,233. After the translation, De
Moivre added a text with numerical illustrations of the main results,
finishing with a brief defense of the design argument.666See De Moivre,, 1738,
p. 243.

Dale, (1988) and Edwards, (1986) argue that in the Approximatio De Moivre
revealed his intention to use Bernoulli’s theorem to solve the inverse
problem. Edwards, (1986) mentions other elements to confirm this hypothesis,
the most compelling one being the Advertisement of the second edition,
published in the first pages of the book. In it De Moivre gives a list of
reasons showing why the second edition was better than the first one. The 13th
reason is that the second edition brings

The Solution of a Question which cannot fail of interesting the Reader; it is,
What reasonable Conjectures may be derived from Experiments, or what are the
Odds that after a certain number of Experiments have been made concerning the
happening or failing of Events, the Accidents of Contingency will not
afterwards vary from those of Observation beyond certain Limits; which leads
naturally to this Consequence, that altho’ Chance has very great Influence on
some Events, yet that it very little disturbs those which in their natural
Institution were designed to happen according to fixt Laws, Chance vanishing
as it were at long run in respect to the Constancy and Regularity of Order.

The text asks if we can reasonably conjecture based on past experience. The
writer of the advertisement answers that we can, saying that while single
events are unpredictable, there is stability and order in the long run. Thus,
the advertisement says the 1738 edition provided an answer to the inverse
problem.777In 1736 the Bibliothèque raisonnée des ouvrages des savans de
l’Europe, a French journal that had a section with reports on new publications
coming out of London, advertised the second edition of The Doctrine of Chances
with almost exactly the same text. Compare the paragraph given above with the
following one, from the Bibliothèque raisonnée, volume 16, p. $218-219$:
“Enfin, on y trouvera la solution de cette Question curieuse & interessante:
Qu’elles conjectures peut-on raisonnablement sonder sur l’Expérience? ou,
Qu’elle probabilité y a-t-il qu’après avoir trouvé par un certain nombre
d’expériences, que des événemens sont ou ne sont pas arrivez les accidens de
la Contingence ne s’éloigneront pas après cela de ceux qu’on a observez au
delà de certaines limites. Ce qui conduit naturellement à cette conséquence;
Que quoi que le Hazard influe beaucoup sur certains evénemens cependant il
dérange fort peu ceux qui ont été d’abord déterminez à arriver selon des Loix
fixes & constantes; le Hazard s’évanouissiant pour ainsi dire, à la longue,
par rapport à la stabilité & au retour de l’ordre.” . Bellhouse, (2011)
conjectured that the author of the 1736 text was Pierre Des Maizeaux. He was
one of the active writers for the Bibliothèque raisonnée. Notwithstanding, the
similitude of both texts indicate that they were probably written by De Moivre
himself. In the 1738 text De Moivre used the first person. See De Moivre,,
1738, p. xii$-$xiv.

###  2.1 David Hartley

Another indication that someone was searching for the solution to the inverse
problem before 1763 is a paragraph from David Hartley’s book, Observations on
Man, His Frame, His Duty, and His Expectations, published in 1749. In a
section where he discusses the refinements to the law of large numbers made by
De Moivre, Hartley added a now famous paragraph.

An ingenious Friend has communicated to me a Solution of the inverse Problem,
in which he has shewn what the Expectation is, when an Event has happened $p$
times, and failed $q$ times, that the original Ratio of the Causes for the
Happening or Failing of an Event should deviate in any given Degree from that
of $p$ to $q$. And it appears from this Solution, that where the Number of
Trials is very great, the Deviation must be inconsiderable: Which shews that
we may hope to determine the Proportions, and, by degrees, the whole Nature,
of unknown Causes, by a sufficient Observation of their Effects.

The resemblance between the mentioned paragraph and the opening lines on
Bayes’s Essay is evident:

Given the number of times in which an unknown event has happened and failed:
Required the chance that the probability of its happening in a single trial
lies somewhere between any two degrees of probability that can be named.

Daston,, 1988, p. 256 and Dale,, 2003, p. 331 comment on the resemblance, the
later mentioning that Hartley used the term expectation, on which Bayes’s
definition of probability was based on, following the classic approach
tradition of the early probabilists Pascal and Huygens.

The paragraph from Hartley, (1749) led historians of probability and
statistics to search for a personal connection between David Hartley and
Thomas Bayes.888The first to conjecture about the connection was probably
Singer, (1979). See also Stigler, (1983, 2013); Dale, (1986, 2003). After the
discovery of some of Bayes’s unpublished manuscripts made by Dale, (1986), the
connection was viewed as plausible, but no conclusive evidence was presented.
See also Bellhouse, (2002, 2007) and Stigler, (2013) with respect to this
possible connection. In Allen, (1999), Richard Allen, Hartley’s biographer,
also conjectures about a possible relationship, but did not present a
conclusive answer.

Bayes and Hartley were both fellows of the Royal Society, Hartley elected in
1736 and Bayes in 1742, but apparently could not be considered individuals
with mutual interests. Hartley was a physician, son of an Anglican clergyman,
while Bayes was a Nonconformist, a dissenting minister with an interest in
mathematics. However, looking closely, it seems quite plausible that they
developed some sort of connection. Their biographies (Allen,, 1999;
Bellhouse,, 2004; Dale,, 2003) show their shared common interests, such as
mathematical and theological subjects, and acquaintances.999William Whiston
($1667-1752$) and Isaac Maddox ($1697-1759$) to mention only two. Whiston,
former Lucasian professor of mathematics was expelled from Cambridge in 1710
due to his religious views. See Allen,, 1999, p. 35 for the relationship of
Whiston and Hartley and Dale,, 2003, p. 83 for his meeting with Bayes.
Originally a dissenter, Maddox studied theology at the University of Edinburgh
(Haydon,, 2009), where he and Bayes attended the logic and metaphysics course
taught by Colin Drummond (c. $1685-1753$) in 1719. See the Matriculation Album
of the University of Edinburgh. Names under the inscription Discipuli Domini
Colini Drummond qui vigesimo-septimo die Februarij, MDCCXIX Subscripserunt and
Maddox, (1750) for his friendship with Hartley. Therefore, for the purposes of
this work, we will assume that Bayes was indeed Hartley’s “ingenious Friend”.

We have been able to establish an early connection between Hartley and Bayes,
albeit a brief one. The Royal Society of London, of which both Bayes and
Hartley were fellows, had weekly meetings to discuss scientific reports from
its members and others that were considered interesting. The minutes of these
meetings do not have a list with all those present, only of those who were not
members and were invited by other fellows. Those who brought some subject or
text to be discussed were also mentioned.

It was usual for men interested in being elected fellows to attend at least
one meeting under the sponsorship of another fellow, so that the candidate
could be introduced to the other members of the Society. On folio 380 of
volume 17 of the Journal Books we read that Bayes attended a meeting of the
Society on March 25, 1742, brought by John Belchier, a surgeon at Guy’s
Hospital in London. The minutes of the same meeting continue on folio 381,
where we find the following entry.

Doctor Hartley presented a pamphlet intitled, An account of the success of Mrs
Stephens’s medicines for the stone in the case of James Kirkpatrick Doctor of
Divinity & M.D. printed at Belfast in Ireland, 1739. For which he had thanks.

David Hartley was a faithful advocate of Mrs. Stephens’s medicine to treat
bladder stones, having published several reports of successful cases of its
use, including the one mentioned above.101010See Allen, (1999) for more on
Hartley and the medicine for bladder stones. These minutes give us, so far,
the only known mention of an occasion when Thomas Bayes and David Hartley were
both present.

But why Hartley did not mention Bayes explicitly? Judging from his notes,
letters and published works, we may say that Bayes was a quiet man who wanted
to avoid quarrels. In 1747 he sent a letter to Philip Stanhope ($1714-1786$),
second Earl Stanhope, who studied mathematics under De Moivre.111111Stanhope
was one of the fellows who proposed Bayes’s election to the Royal Society in
1742. The subject was a series firstly derived by James Stirling ($1692-1770$)
and used by De Moivre in the demonstration of the normal approximation to
binomial probabilities. De Moivre believed the series converged, providing an
approximation to $\sqrt{2\pi}$. In the letter to Stanhope, Bayes showed the
series diverges. He started the communication with “It has been asserted by
some eminent mathematicians”, avoiding the reference of De Moivre and Stirling
since he knew Stanhope would certainly inform the Frenchman about the mistake.
The letter was published in the Philosophical Transactions, after Bayes’s
death.121212Bayes, 1763b .

He published only two pamphlets in his lifetime, both anonymously, both on
controversial subjects: Bayes, (1731) on divine benevolence, replying the
views of John Balguy ($1686-1748$), an Anglican theologian, and Bayes, (1736)
on differential calculus, defending Newton’s approach against the criticism
made by George Berkley ($1685-1753$), the bishop of Cloyne, in The Analyst
(1734). Although his contemporaries knew that he was the author of both
works,131313See Doddridge, (1763) and De Morgan, (1860). One of Bayes’s
obituaries says that he was the author of a well received pamphlet on divine
benevolence. See The Library Or Moral and Critical Magazine, volume 1, May
1761, p.110. they would not mention his authorship. Even Price, (1758), the
first published book of his friend Richard Price, mentions Bayes, (1731) but
not the name of the author. Only the third edition of Price’s book, published
in 1787, mentions Bayes explicitly. Thus, a plausible hypothesis is that Bayes
asked Hartley to not be mentioned.

##  3 Bayes and Hume

We think Bayes’s work on probability was not motivated as a response to Hume,
(1748). Firstly because in his publications and notes Bayes never sought
applications, practical or philosophical, for the mathematical results he
derived. Secondly because, if Bayes communicated his findings to Hartley as we
believe, the chronologies of the publications of Hume, (1748) and Hartley,
(1749) indicate that it is unlikely that Bayes wanted to refute Hume.

Hume’s book with the essay on miracles was advertised in April of
1748.141414See (Hume,, 2000, p. xxxvi), the Gentleman’s Magazine, volume 18,
page 192, April of 1748 and The London Magazine, volume 17, page 240, May of
1748. Stigler, (1983) says that Hartley’s book was substantially complete by
1739 and fully finished by 1745 but Webb, (1988) reports that Hartley prepared
a new draft of Observations on Man that was completed only in the winter of
1747. This draft was sent to some of his friends for corrections and useful
remarks, and perhaps that was the time when Hartley’s “ingenious Friend”
communicated his discovery. Corroborating this hypothesis, notice that the
dates found on Bayes’s notebook151515Dale, (1986). show that he was working on
the Essay circa $1747-1750$.

The final version of Hartley, (1749) was sent to the press in the beginning of
December of 1748, and a presentation copy was sent to the Royal Society in
February of 1749.161616Journal Book Meetings, volume 21, folio 65, The Royal
Society of London. See also the Gentleman’s Magazine, volume 19, page 96,
February of 1749. It is impossible to know when Hartley wrote the paragraph
about the inverse problem but, if Bayes was Hartley’s friend and wanted to
refute Hume, he had no more than eight months to read Hume, (1748), think
about a reply, solve the problem and communicate the solution to Hartley,
which seem unlikely.171717On the other hand, since Hume’s main point on
induction was already present in Hume, (1739) and Hume, (1740), it is possible
that Bayes was already thinking about a reply before 1748 and all the debate
about miracles raised by the Enquiry just gave him the final incentive to work
on a rebuttal. Favoring the hypothesis that Bayes wanted to refute Hume’s
argument, Stigler, (2013) found offprints of Bayes’s essay with a different
title: “A Method of Calculating the Exact Probability of All Conclusions
founded on Induction”. A follow up of Bayes, 1763a , Price, (1764), published
with the title “A Demonstration of the Second Rule in the Essay toward the
Solution of a Problem in the Doctrine of Chances”, also has an offprint with a
different title, “A Supplement to the Essay on a Method of Calculating the
Exact Probability of All Conclusions founded on Induction”. Since the
offprints were printed at the author’s expense, Stigler, (2013) says that
Price almost certainly chose these titles, which are more informative and
support better the case against Hume. This led Stigler to claim that it is not
improbable that Bayes saw his work as a defensive tool to combat Hume, like
Price did.

##  4 Bayes and De Moivre

Bellhouse, (2011) shows that Bayes did not study probability under De Moivre.
However, we can affirm that Bayes knew The Doctrine of Chances: letters
exchanged between Bayes and Stanhope in the late 1740’s181818See Bellhouse,
(2007) and Bellhouse,, 2011, ch. 13. show them discussing, among other
mathematical subjects, probability problems found in the De Moivre, (1738).

One topic of discussion in their correspondence was the problem of runs: given
a sequence of $n\in\mathbb{N}$ independent Bernoulli trials all with the same
probability of success, what is the probability of a run or streak of $r\leq
n$, $r\in\mathbb{N}$, successes in a row?191919Problem LXXXVIII of De Moivre,
(1738). It is worth noticing that this is the first problem discussed after
the translation of the Approximatio given in De Moivre, (1738).202020See De
Moivre,, 1738, p. 243.

Another topic discussed by Bayes and Stanhope was mentioned above: the series
used to derive the normal approximation to binomial probabilities. The result
was published in the Approximatio and in De Moivre, (1738). In a manuscript
sent to Stanhope, Bayes used a different argument to derive his own
approximation to the factorial of a large positive integer, and then applied
it to approximate the ratio of the middle term in the expansion of $(1+1)^{n}$
to $2^{n}$, ultimately obtaining the same approximation found by De Moivre. If
Stanhope informed De Moivre about Bayes’s work, it would probably have annoyed
the Frenchman. Bayes’s demonstration was an indication of an error, even
though minor, but in a major discovery.

Hence, the Bayes$-$Stanhope correspondence shows that they were studying the
final part of De Moivre, (1738) circa $1745-47$. Bayes’s notebook corroborates
this hypothesis since it has notes on probability made between $1747$ and
$1750$ with results published in his Essay.212121See Dale, (1986, 2003) Our
conjecture is that Bayes, while studying the Approximatio thought about a
different approach to find inverse probabilities. He eventually found an
innovative way sometime before the end of 1748, being able to communicate the
main point of his finding to Hartley.

##  5 Bayes and Murdoch

The third edition of The Doctrine of Chances was published in 1756, two years
after De Moivre’s death. In the advertisement for the 1756 edition we read
that De Moivre, due to his feeble health, trusted the new edition to “one of
his Friends; to whom he gave a Copy of the former [1738 edition], with some
marginal Corrections and Additions, in his own hand writing. To these the
Editor has added a few more, where they were thought necessary.”

The mentioned friend was Patrick Murdoch, a Scottish Anglican clergyman who
was also recognized as a good mathematician. In 1723 he graduated from the
University of Edinburgh, where he was contemporary of Thomas Bayes, Nathaniel
Carpenter (Bayes’s cousin) and Isaac Maddox, who became bishop of Worcester
and friend of David Hartley. Murdoch, Maddox and Bayes attended the same logic
course taught by Colin Drummond in 1719.222222Matriculation Album of the
University of Edinburgh.

De Moivre probably chose Murdoch to edit his book because the clergyman had a
successful experience editing Maclaurin’s papers after his death. The
editorial work resulted in An Account of Sir Isaac Newton’s Philosophical
Discoveries, In Four Books published in $1748$ with a prefatory biography of
Maclaurin written by Murdoch. Bellhouse, (2011) shows that Murdoch asked De
Moivre’s friends and former pupils, like Stanhope, to send unpublished
material to be added in an appendix to the third edition, without naming the
authors. An analysis of Stanhope’s unpublished works on probability shows that
it is unlikely that he submitted anything to Murdoch. Also according to
Bellhouse, (2011), it is impossible to know which parts of the 1756 edition
were added by Murdoch, Henry Stewart Stevens (d. 1760) and George Lewis Scott
($1708-1780$), who all helped to edit the book.

However, some parts of the added text can be plausibly attributed to hands
that were not De Moivre’s. One of them is a remark added after the translation
of the Approximatio. The text after Corollary 10, already present in the 1738
edition, was named Remark I and ended defending the design argument. The text
that follows is a more extensive defense of the same argument. Named as Remark
II, it is almost three pages long, and starts with:232323De Moivre,, 1756, p.
251.

As, upon the Supposition of a certain determinate Law according to which any
Event is to happen, we demonstrate that the Ratio of Happenings will
continually approach to that Law, as the Experiments or Observations are
multiplied: so, conversely, if from numberless Observations we find the Ratio
of the Events to converge to a determinate quantity, as to the Ratio of $P$ to
$Q$; then we conclude that this Ratio expresses the determinate Law according
to which the Event is to happen.

The text basically states that De Moivre’s result given in the Approximatio
would also provide a proof for the inverse problem, if we could perform
“numberless” or an infinite number of observations. The following paragraph
argues to prove the statement by contradiction.

The similarity between the opening paragraph of Remark II and Hartley’s
comment on his friend’s discovery is evident242424See the paragraph from
Hartley, (1749) quoted on section 2.1. but, so far, the chronology of the
texts remained unnoticed or was misinterpreted.252525Daston, (1988) says that
Hartley “did not consider De Moivre’s ‘remark’ a sufficient guarantee of the
inverse theorem, for he went on to mention an unnamed ‘ingenious Friend’ who
had shown him the solution to the ‘inverse Problem’ […]” Since Remark II was
not in the 1738 edition and Hartley’s book was published in 1749, it is
certain that Hartley did not see De Moivre’s (or his editors’) argument about
the inverse problem. The influence was, most probably, the opposite: Murdoch
read Hartley’s book and basically reproduced the point made by his “ingenious
Friend”, crediting De Moivre.262626Hartley was no mathematician but had a
solid background on the subject, being able to understand and apply
mathematical tools in his works, like Hartley, (1733), a pamphlet on
inoculation. However, it is unlikely that he collaborated with De Moivre on
any mathematical work, since there is no surviving evidence of such a
partnership. Murdoch’s library was sold in 1777 with the libraries of four
other people.272727Zachary Pearce ($1690-1774$), bishop of Bangor and F.R.S.,
William Lowndes ($1652-1724$), English politician, and two other reverends,
Pawlet St. John and Barshaw. The catalogue of the libraries lists one copy of
Hartley, (1749), although it does not mention whether or not it belonged to
Murdoch.

There are other elements allowing us to credit Murdoch for the authorship of
Remark II. Firstly, to reinforce his point on design, the author mentions a
probabilistic argument for divine providence from the ratio of male to female
births, made by the Scottish physician John Arbuthnot ($1667-1735$) in
1710.282828John Arbuthnot (1710). “An argument for Divine Providence, taken
from the constant regularity observed in the births of both sexes”.
Philosophical Transactions of the Royal Society of London, 27, p. 186–190.
Arbuthnot is described in De Moivre, (1756) as an “excellent Person,” which,
based on an earlier comment by De Moivre, may seem strange.

In the preface of the The Doctrine of Chances written in 1717, De Moivre
remembers the time he read Huygens’s De Ratiocinnis in Ludo Aleae (1657) and
its English translation “done by a very ingenious Gentleman, who, tho’ capable
of carrying the matter a great deal farther, was contented to follow his
Original; adding […] some few things more.” The mentioned translation of
Huygens, anonymously published in 1692, was assigned to Arbuthnot. De Moivre’s
reference is simultaneously polite and ironic, perhaps showing his resentment
of the Scottish group which Arbuthnot was part of.292929De Moivre was involved
in a rough dispute with another Scottish physician, George Cheyne
($1672-1743$), friend of Arbuthnot and David Gregory ($1666-1708$). In a
letter to Johann Bernoulli, De Moivre says of Gregory that “It seems to me
that he is suffering from a malady common to the Scotch.” See Walker, (1934).

A second piece of evidence pointing to Murdoch’s authorship is a response to
David Hume ($1711-1776$) present in the text of Remark II. Murdoch and Hume
were close friends of Andrew Millar ($1705-1768)$, a Scottish publisher of
several important British authors of the eighteenth century, including De
Moivre and Hume. Section VI of Hume, (1748), the essay Of Probability, starts
with the following sentence.303030See Hume,, 1748, p. 93.

Tho’ there be no such Thing as Chance in the World; our Ignorance of the real
Cause of any Event has the same Influence on the Understanding, and begets a
like Species of Belief or Opinion.

Moreover, the essay Of Liberty and Necessity has the sentence
below.313131Hume,, 1748, Sec. VIII, p .151.

’Tis universally allow’d, that nothing exists without a Cause of its
Existence, and that Chance, when strictly examined, is a mere negative Word,
and means not any real Power, which has any where, a Being in Nature.

To those statements, the author of Remark II replied with the
following.323232Hacking,, 1990, p. 13 points out these references but
incorrectly identifies the text as from De Moivre, (1738), when in fact it was
available only in the 1756 edition. This mistake led Hacking to assume that
Hume, (1748) was referring to De Moivre’s epithet, that chance is a mere word,
when the actual timing, and possible influence or reference, was the opposite.

Chance, as we understand it, supposes the Existence of things, and their
general known Properties: that a number of Dice, for instance, being thrown,
each of them shall settle upon one or other of its Bases. After which, the
Probability of an assigned Chance, that is of some particular disposition of
the Dice, becomes as proper a subject of Investigation as any other quantity
or Ratio can be.

But Chance, in atheistical writings or discourse, is a sound utterly
insignificant: It imports no determination to any mode of Existence; nor
indeed to Existence itself, more than to non-existence; it can neither be
defined nor understood: nor can any Proposition concerning it be either
affirmed or denied, excepting this one, ‘That it is a mere word.’

Therefore, considering the length of Remark II, its arduous defense of the
design argument with complimentary references to Arbuthnot and Nicholas
Bernoulli and a rebuttal of a sceptical philosopher,333333Interestingly, Hume
used at least three passages from Maclaurin, (1748), edited by Murdoch, in his
works. Compare Hume,, 1748, p. 56 with Maclaurin,, 1748, p. 105 and Hume,,
1779, p. 42-44,45-46 with Maclaurin,, 1748, p.381,378-9. Hume’s library had a
copy of Maclaurin, (1748), as showed by Norton and Norton, (1996). we can
safely conclude that the text was not penned by De Moivre. On the other hand,
it is plausible to assume it coming from a clergyman like Murdoch.

Finally, the last paragraph of the remark says that Bernoulli’s result can be
applied to several fields, as shown in Part IV of Bernoulli, (1713) and the
last sentence of the section is the following.

Yet there are Writers, of a Class indeed very different from that of James
Bernoulli, who insinuate as if the Doctrine of Probabilities could have no
place in any serious Enquiry; and that Studies of this kind, trivial and easy
as they be, rather disqualify a man for reasoning on every other subject. Let
the Reader chuse.

This may be a reference to the closing paragraph of Bayes, (1736), the
pamphlet on differential calculus.

To conclude; as I would not be thought, by any thing I have said, to be an
enemy to true Logic and sound Metaphysics; and on the contrary think the most
general use of the Mathematics is to inure us to a just way of thinking and
arguing; it is a proper inquiry, I imagine, for those who have the direction
of the education of youth, in what manner mathematical studies may be so
pursued as most surely to answer this end […] for so far as Mathematics do not
tend to make men more sober and rational thinkers, wiser and better men, they
are only to be considered as an amusement, which ought not to take us off from
serious business.

The text is probably informing Bayes’s opinion on the purpose of teaching pure
mathematics in dissenting academies. Bellhouse, (2010) describes the
pedagogical reasons that justified the teaching of mathematics in those
academies.343434Consider for instance the view of Philip Doddridge
($1702-1751$), a dissenting educator: he saw mathematics not only as a
precursor to the study of the new Newtonian science, but also as a tool to
help mankind to better understand how God works in creation.

We do not have much evidence to judge Murdoch’s relationship with Bayes, but a
letter sent by Bayes to Stanhope in 1755 reveals something.353535See
Bellhouse, (2002). The letter shows Bayes commenting on an article written by
Murdoch, probably at Stanhope’s request, and the comments were not favourable.
If Bayes and Murdoch were close friends, the latter would ask directly for
comments on his work, not waiting for someone else to be an intermediary in
the request.

Therefore, we think it is plausible to assume that sometime between 1747 and
1755, Stanhope informed De Moivre and/or Murdoch about Bayes’s results on
inverse probability, without providing much detail. De Moivre did not like it
and Murdoch, asked by De Moivre or by his own initiative, wrote Remark II to
claim De Moivre’s priority on the solution and to reply to both Hume and
Bayes. Considering Remark II on its context, we realized there are no
significant reasons for the text to be there: it does not add relevant theory
or philosophical points (Remark I already made them) and it is unrelated to
the problems following the text. Given this, we may reasonably conjecture
that, in fact, Murdoch knew that Bayes was Hartley’s ingenious friend, but
considered De Moivre to be the first to have solved the inverse problem.
Richard Price, Bayes’s friend and intellectual executor disagreed.

##  6 Concluding remarks

After Thomas Bayes’s death his family asked Price to look into his papers,
among which Price found notes which he edited and expanded in an appendix with
numerical illustrations. The final result, named An Essay Towards Solving a
Problem in the Doctrine of Chances, was sent with a cover letter to John
Canton ($1718-1772$), a fellow of the Royal Society and common friend.

The first remark made by Price is that Bayes realized he needed what we now
call a prior distribution in order to solve the inverse problem, something De
Moivre and Jakob Bernoulli did not think of. It is also clear from the text
that Price rejected the alleged proof of the inverse problem presented in
Remark II of De Moivre, (1756),363636Price used the 1756 edition as reference,
so he knew Remark II. Price mentioned that Bayes deduced the solution of the
“converse problem”, echoing De Moivre,, 1756, p. 251 where we read, “so,
conversely”. recognizing Bayes as the first to solve it.

It may be safely added, I fancy, that it is also a problem that has never
before been solved. Mr. De Moivre, […] has […], after Bernoulli,[…], given
rules to find the probability there is, that if a very great number of trials
be made concerning any event, the proportion of the number of times it will
happen, to the number of times it will fail in those trials, should differ
less than by small assigned limits from the proportion of the probability of
its happening to the probability of its failing in one single trial. But I
know of no person who has shewn how to deduce the solution of the converse
problem to this; namely, “the number of times an unknown event has happened
and failed being given, to find the chance that the probability of its
happening should lie somewhere between any two named degrees of probability.”
What Mr. De Moivre has done therefore cannot be thought sufficient to make the
consideration of this point unnecessary: especially, as the rules he has given
are not pretended to be rigorously exact, except on supposition that the
number of trials made are infinite; from whence it is not obvious how large
the number of trials must be in order to make them exact enough to be depended
on in practice.

Price recognized De Moivre’s work refining Bernoulli’s result, but saw Remark
II as useless in practice, since it relies on the “supposition that the number
of trials made are infinite.” Possibly defending his friend against the
criticism of Murdoch, Price underlined the importance of the problem solved by
Bayes.

Every judicious person will be sensible that the problem now mentioned is by
no means merely a curious speculation in the doctrine of chances, but
necessary to be solved in order to a sure foundation for all our reasonings
concerning past facts, and what is likely to be hereafter.[…]

These observations prove that the problem enquired after in this essay is no
less important than it is curious.

The possibility of a defense of Bayes, (1736) gains weight when Price
explicitly mentions Murdoch’s criticism against it.

Mr. De Moivre calls the problem he has thus solved, the hardest that can be
proposed on the subject of chance. His solution he has applied to a very
important purpose, and thereby shewn that those a remuch mistaken who have
insinuated that the Doctrine of Chances in mathematics is of trivial
consequence, and cannot have a place in any serious enquiry. The purpose I
mean is […] to confirm the argument taken from final causes for the existence
of the Deity. It will be easy to see that the converse problem solved in this
essay is more directly applicable to this purpose […]

In a very polite way Price acknowledged De Moivre’s (or Murdoch’s?) efforts in
trying to prove the design argument from probabilistic arguments but points
out that Bayes’s solution is more appropriate to this end.

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## Appendix: Bernoulli’s theorem and its inverse

The objective of this appendix is to formally differentiate Bayes’s theorem
from the inverse use of Bernoulli’s law of large numbers. In order to do this
we will use modern notation. If $\\{X_{i}\\}_{i=1}^{n}$, $n\in\mathbb{N}$, are
independent Bernoulli random variables such that
$E(X_{i})=P(X_{i}=1)=\theta\in[0,1]$, then for any given real number
$\varepsilon>0$, Jakob Bernoulli proved that:

$\lim_{n\to\infty}P(\theta-\varepsilon<\overline{X}_{n}<\theta+\varepsilon\mid\theta)=1,$

where $\overline{X}_{n}=\sum_{i=1}^{n}X_{i}/n$. Notice that Bernoulli assumed
that $\theta$ and $\varepsilon$ were known to compute $n$ such that
$P(\theta-\varepsilon<\overline{X}_{n}<\theta+\varepsilon\mid\theta)$ would be
as close to 1 as desired.373737 As a numerical example, Bernoulli,, 1713, p.
238 found $n=25,550$ for $\theta=3/5$ and $\varepsilon=1/50$ in order to make
$P(|\theta-\overline{X}_{n}|<\varepsilon)=0.999$. Thus, the question Bernoulli
answered was: assuming that the probability of an event is known, how many
times an experiment ought to be repeated before one can be “morally” certain
that the relative frequency of the event is close enough to the true
probability?

If the event probability ($\theta$) is unknown and a given number of successes
and failures is obtained after $n$ trials, one can use Bernoulli’s law
inversely by taking the relative frequency of successes as an approximation to
the true but unknown probability. Although Dale, (1988), Hacking, (1975) and
Todhunter, (1865) argue that Bernoulli had this use in mind, this view cannot
be proven.

On his side, Bayes wanted to solve the inverse problem as stated in Section 2:
his goal was to compute for any given limits $\ell_{1}$ and $\ell_{2}$,
$P(\ell_{1}\leq\theta\leq\ell_{2}~{}|~{}\overline{x}_{n})$, where
$\overline{x}_{n}$ is the observed value of $\overline{X}_{n}$. In fact,
Bayes’s result is not a limit theorem and, as such, cannot be formally seen as
the inverse of Bernoulli’s law. Laplace, (1774) eventually proved the inverse
of Bernoulli’s law based on inverse probability and his method of asymptotic
approximation for integrals.

As put by Gorroochurn,, 2012, p. 140, the difference between Bernoulli’s law,
the inverse use of Bernoulli’s law and Bayes’s theorem may be summarized as:

  * •

Bernoulli’s law: for given $\theta$ and $\varepsilon>0,$ we can find
$n\in\mathbb{N}$ such that
$P(\theta-\varepsilon<\overline{X}_{n}<\theta+\varepsilon\mid\theta)$ is
arbitrarily close to 1;

  * •

inverse use of Bernoulli’s law: when $\theta$ is unknown, given
$\overline{x}_{n}$,
$P\left(\overline{x}_{n}-\varepsilon<\theta<\overline{x}_{n}+\varepsilon\right)$
is close to $1$, when $n$ is large, so that $\overline{x}_{n}\approx\theta$.

  * •

Bayes’s theorem: when $\theta$ is unknown, for given
$\ell_{1},\ell_{2},\overline{x}_{n}$ and assuming that $\theta$ has some prior
distribution, then we can calculate
$P\left(\ell_{1}<\theta<\ell_{2}\mid\overline{x}_{n}\right)$.

Under this perspective, one may say that Bayes’s approach was the “correct”
one to solve the inverse problem as considered in this text. The inverse use
of Bernoulli’s law by other authors, such as Halley, (1693), did not
foreshadow by any way the method applied by Bayes to compute the required
conditional probability.


# Symmetry defect of $n$\- dimensional complete intersections in
$\mathbb{C}^{2n-1}$

L.R.G. Dias & Z. Jelonek  Faculdade de Matemática, Universidade Federal de
Uberlândia, Av. João Naves de Ávila 2121, 1F-153 - CEP: 38408-100, Uberlândia,
Brasil<EMAIL_ADDRESS>Instytut Matematyczny
Polska Akademia Nauk
Śniadeckich 8, 00-956 Warszawa, Poland<EMAIL_ADDRESS>
###### Abstract.

Let $X,Y\subset\mathbb{C}^{2n-1}$ be $n$-dimensional strong complete
intersections in a general position. In this note, we consider the set of
midpoints of chords connecting a point $x\in X$ to a point $y\in Y$. This set
is defined as the image of the map $\Phi(x,y)=\frac{x+y}{2}.$ Under geometric
conditions on $X$ and $Y$, we prove that the symmetry defect of $X$ and $Y$,
which is the bifurcation set $B(X,Y)$ of the mapping $\Phi$, is an algebraic
variety, characterized by a topological invariant. We introduce a hypersurface
that approximates the set $B(X,Y)$ and we present an estimate for its degree.
Moreover, for any two $n$-dimensional strong complete intersections
$X,Y\subset\mathbb{C}^{2n-1}$ (including the case $X=Y$) we introduce a
generic symmetry defect set $\tilde{B}(X,Y)$ of $X$ and $Y$, which is defined
up to homeomorphism.

###### Key words and phrases:

symmetric defect, global theory of singularities, Wigner caustic.

###### 1991 Mathematics Subject Classification:

MSC Classification: 58 K 30, 14 D 06

##  1\. Introduction

Over the last two decades numerous methods have been developed to study affine
geometry of surfaces and curves especially their affinely invariant symmetry
characteristics. The symmetry sets [2, 3, 4] and the center symmetry sets were
investigated extensively in [6, 7, 8]. Several constructions of the set
equivalent to the point of central symmetry for perturbed centrally symmetric
ovals were presented in the literature and resulted in the kind of symmetry
defect called center symmetry set. The center symmetry set directly appears in
the construction of the so-called Wigner caustic. This caustic is obtained by
the stationary phase method applied to the semiclassical Wigner function which
completely describes a quantum state in the symplectic phase space [1]. It is
built of points where the central symmetry, i.e. the number of intervals
ending in the surface and passing centrally through that point, changes. We
call this set a symmetry defect or bifurcation set. In [9] this construction
was generalized for algebraic varieties $Z^{n}\subset\mathbb{C}^{2n}$.

Let us recall that a pair of points $x,y$ of a plane curve $Z$ is called a
parallel pair if their tangent lines are parallel. Given $\lambda\in[0,1]$,
the affine $\lambda$-equidistant is the set of points $\lambda x+(1-\lambda)y$
such that $x,y$ is a parallel pair of $Z$. In the special case $\lambda=1/2$,
the affine $1/2$-equidistant is mentioned above the Wigner Caustic.

Related to the affine $1/2$-equidistant, it was proved in [9] the following
general result: let $Z\subset\mathbb{C}^{2n}$ be a $n$-dimensional algebraic
variety in a general position. For $a\in\mathbb{C}^{2n}$, let $\mu(a)$ be the
number of chords of $Z$ whose midpoint is $a$. Then there exists an algebraic
set $B(Z)\subset\mathbb{C}^{2n}$, called symmetric defect of $Z$, such that
$\deg B(Z)<(\deg Z)^{2}(1+2n(\deg Z-1))$ and $\mu(a)$ is constant on
$\mathbb{C}^{2n}\setminus B(Z)$.

This note is motivated by the recent results of [5] and [9]. We address the
problem of how to introduce a symmetric defect set associated to
$n$-dimensional strong complete intersections $X,Y\subset\mathbb{C}^{2n-1}$.
We also investigate algebraic and topological properties of this set.

In order to state our theorems properly, we introduce the following
definitions:

######  Definition 1.1.

We say that an $n$-dimensional affine variety $Z\subset\mathbb{C}^{m}$ is a
strong complete intersection, if the following conditions are satisfied:

1) $Z=\\{z\in\mathbb{C}^{m}\mid h(z)=0\mbox{ and
}\mathrm{rank}\,dh(z)=m-n\\}$, for some polynomial mapping
$h=(h_{1},\ldots,h_{m-n})\colon\mathbb{C}^{m}\to\mathbb{C}^{m-n}$.

2) $\dim\\{{(h_{1})}_{*}=0,...,{(h_{m-n})}_{*}=0\\}=m-n$, where ${h_{i}}_{*}$
denotes the highest homogeneous component of $h_{i}.$

If $\deg h_{i}=a_{i}$, then we say that $Z$ has multi-degree
$(a_{1},...,a_{m-n}).$

Now, let $X,Y\subset\mathbb{C}^{2n-1}$ be $n$-dimensional strong complete
intersections, we say:

$X=\\{z\in\mathbb{C}^{2n-1}\mid f(z)=0\mbox{ and
}\mathrm{rank}\,df(z)=n-1\\},$ $Y=\\{z\in\mathbb{C}^{2n-1}\mid g(z)=0\mbox{
and }\mathrm{rank}\,dg(z)=n-1\\},$

with $f=(f_{1},\ldots,f_{n-1})\colon\mathbb{C}^{2n-1}\to\mathbb{C}^{n-1}$ and
$g=(g_{1},\ldots,g_{n-1})\colon\mathbb{C}^{2n-1}\to\mathbb{C}^{n-1}$
polynomial mapping.

######  Definition 1.2.

We say that $X$ and $Y$ are in general positions if the following holds

$\dim\left(\left(\bigcap_{1}^{n-1}\\{(f_{i})_{*}=0\\}\right)\bigcap\left(\bigcap_{1}^{n-1}\\{(g_{i})_{*}=0\\}\right)\right)=1.$

We consider

(1.1) $\Phi:X\times Y\to\mathbb{C}^{2n-1}\mbox{ given by
}\Phi(x,y)=\frac{1}{2}(x+y),$

and we denote by $K_{0}(\Phi)$ the critical values set of $\Phi$. If
$p\in\mathbb{C}^{2n-1}\setminus K_{0}(\Phi)$, we set

$\mu_{X,Y}(p):=\chi(\Phi^{-1}(p)),$

where $\chi(\Phi^{-1}(p))$ denotes the Euler Characteristic of $\Phi^{-1}(p)$.
The bifurcation set of $\Phi$ is denoted by $B(X,Y),$ see Definition 2.3. We
also define the asymptotic part
$B_{\infty}(X,Y)=B(X,Y)\setminus\overline{K_{0}(\Phi)},$ which contains only
bifurcation points of $\Phi$ which comes from infinity. Note that it is not a
standard definition.

As in [9], the set $B(X,Y)$ is also called a symmetric defect of $X$ and $Y$.
The following result is obvious:

######  Proposition 1.3.

Let $X,Y\subset\mathbb{C}^{2n-1}$ be $n$-dimensional strong complete
intersections. Then, for a generic linear mapping
$H:\mathbb{C}^{2n-1}\to\mathbb{C}^{2n-1}$, the sets $X$ and $H(Y)$ are in a
general position.

In this note, we prove the following results related to the symmetric defect:

######  Theorem 1.4.

Let $X^{n},Y^{n}\subset\mathbb{C}^{2n-1}$ be strong complete intersections in
general positions. There exists a number $\mu_{X,Y}$ such that, for any
$p\notin\overline{K_{0}(\Phi)}$, we have $p\notin B_{\infty}(X,Y)$ if and only
if $\mu_{X,Y}=\mu_{X,Y}(p)$. Moreover, the set $B(X,Y)$ is an algebraic set.

######  Theorem 1.5.

Let $X^{n},Y^{n}\subset\mathbb{C}^{2n-1}$ be strong complete intersections in
general position of multi-degrees $(a_{1},...,a_{n-1})$ and
$(b_{1},...,b_{n-1})$, respectively. Then $B_{\infty}(X,Y)$ is contained in a
hypersurface $L_{\infty}(\Phi)$ such that

$\deg L_{\infty}(\Phi)\leq D+d-\mu_{X,Y},$

where $D:=\deg\overline{\mathrm{Sing}(\Phi,L)\setminus\mathrm{Sing}(\Phi)}$,
$d:=\mu(\Phi,L)$, $\mu(\Phi,L)$ is the geometric degree of $(\Phi,L)$ and
$L:\mathbb{C}^{2n-1}\to\mathbb{C}$ denotes a generic linear mapping. In
particular,

$\deg L_{\infty}(\Phi)\leq
D-1\leq\prod^{n-1}_{i=1}a_{i}\prod^{n-1}_{i=1}b_{i}(\sum^{n-1}_{i=1}a_{i}+b_{i}-2)-1.$

Also, we have the following generic case:

######  Theorem 1.6.

Let $X^{n},Y^{n}\subset\mathbb{C}^{2n-1}$ be strong complete intersections.
For a generic linear mapping $H:\mathbb{C}^{2n-1}\to\mathbb{C}^{2n-1}$, the
sets $X^{n}$ and $H(Y^{n})$ are in general positions and the number
$\mu_{X,H(Y)}$ does not depend of $H$. Moreover, for generic $H$ all sets
$B(X,H(Y))$ are ambient homeomorphic, i.e, pairs
$(\mathbb{C}^{2n-1},B(X,H(Y)))$ and $(\mathbb{C}^{2n-1},B(X,H^{\prime}(Y)))$
are homeomorphic for generic $H,H^{\prime}.$

This allows us to define the set $\tilde{B}(X,Y):=B(X,H(Y))$ (where $H$ is
generic) for every two strong complete intersections
$X^{n},Y^{n}\subset\mathbb{C}^{2n-1}$ (including the case $X=Y$) at least up
to homeomorphism. It may happen that $\tilde{B}(X,Y)\not\cong B(X,Y)$.

Theorem 1.4 and 1.5 are proved in §2.2 and Theorem 1.6 in §2.3.

We point out that a version of Theorem 1.4 was presented in [16] for a class
of polynomial functions from $\mathbb{C}^{n}$ to $\mathbb{C}^{n-1}$ and in
[13] for morphisms from $X$ to $Y$, with $X,Y$ affine varieties and $\dim
X=\dim Y+1$. However, the algebraicity of the bifurcation set was not
discussed in those papers.

##  2\. Proofs of the results

###  2.1. Non-properness set

We recall some results on the non-properness set of a polynomial mapping. We
follow [9, 10, 11]. Let $Z_{1},Z_{2}$ be affine varieties over $\mathbb{C}$.

######  Definition 2.1.

Let $f:Z_{1}\rightarrow Z_{2}$ be a generically finite (i.e. a generic fiber
is finite) and dominant polynomial mapping (i.e. $\overline{f(Z_{1})}=Z_{2}$).
We say that $f$ is proper at $y\in Z_{2},$ if there exists an open
neighborhood $U$ of $y$ such that the mapping
$f\mid_{f^{-1}(U)}:f^{-1}(U)\rightarrow U$ is a proper mapping.

We denote by $S_{f}$ the set of points at which the mapping $f$ is not proper.
The set $S_{f}$ is called the non-properness set of $f.$

Let $Z\subset\mathbb{C}^{m}$ be an affine variety of codimension $k$. The
degree of $Z$ is the number of common points of $Z$ and sufficiently general
linear subspace $V\subset\mathbb{C}^{m}$ of dimension $k$. The degree of $Z$
is denoted by $\deg Z$. We have $\deg\mathbb{C}^{m}=1$, for any
$m\in\mathbb{N}$.

The following result is useful (see for instance [9, 10, 11]):

######  Theorem 2.2.

Let $f=(f_{1},...,f_{n}):Z\to\mathbb{C}^{n}$ be a generically finite dominant
polynomial mapping, where $Z\subset\mathbb{C}^{m}$ is an affine
$n$-dimensional variety with $\deg Z=D$. Then $S_{f}$ is either empty or a
hypersurface with

${\rm deg}\ S_{f}\leq\frac{D(\prod^{n}_{i=1}{\rm deg}\
f_{i})-\mu(f)}{\min_{1\leq i\leq n}\deg\ f_{i}},$

where $\mu(f)=(\mathbb{C}(X):\mathbb{C}(f_{1},...,f_{n}))$ is the geometric
degree of $f.$

Theorems 1.4, 1.5 and 1.6 are related with the following definition:

######  Definition 2.3.

Let $f:Z_{1}\to Z_{2}$ be a polynomial mapping. The bifurcation set of $f$,
denoted by $B(f)$, is the smallest subset of $Z_{2}$ such that the restriction
$f_{|}:Z_{1}\setminus f^{-1}(B(f))\to Z_{2}\setminus B(f)$ is a $C^{\infty}$
locally trivial fibration.

Let $Z_{1},Z_{2}$ be smooth $n$-dimensional varieties over $\mathbb{C}$ and
$f:Z_{1}\to Z_{2}$ a dominant polynomial mapping. We denote by $\mu(f)$ the
geometric degree of $f$.

The following useful result follows from the proof of Theorem 2.4 of [9]:

######  Theorem 2.4 ([9, Theorem 2.4]).

Let $Z$ be a smooth affine complex variety of dimension $n.$ Let
$f:Z\to\mathbb{C}^{n}$ be a dominant polynomial mapping. Then,

$B(f)=K_{0}(f)\cup S_{f},$

and

$B(f)=\\{y\in\mathbb{C}^{n}:\\#f^{-1}(y)\not=\mu(f)\\}.$

Moreover, $B(f)$ is either empty (and so $f$ is an isomorphism) or a closed
hypersurface.

The next results are used in the proofs of Theorems 1.4 and 1.5:

######  Theorem 2.5.

Let $X,Y$ be complex irreducible algebraic varieties and $f:X\to Y$ a
polynomial mapping. Let $W_{1},...,W_{r}\subset X$ be closed subvarieties of
$X.$ Then, there exists a nonempty Zariski open subset $U\subset Y$ such that,
for every $y_{1},y_{2}\in U$, the $r+1$-tuples $(f^{-1}(y_{1}),W_{i}\cap
f^{-1}(y_{1});i=1,...,r)$ and $(f^{-1}(y_{2}),W_{i}\cap
f^{-1}(y_{2}),i=1,...,r)$ are homeomorphic.

###### Proof.

We may assume that $Y$ is smooth and irreducible. Let $X_{1}$ be an algebraic
completion of $X.$ Take $X_{2}:=\overline{graph(f)}\subset X_{1}\times{Y}.$

We can assume that $X\subset X_{2}.$ We have an induced mapping
$\overline{f}:X_{2}\to{Y}$ such that $\overline{f}_{|X}=f.$ Let
$Z=X_{2}\setminus X.$ Denote by $\overline{W_{i}},$ for $i=1,...,r$, the
closures of $W_{i}$ in $X_{2}.$ Let $\mathcal{R}=\\{Z,\overline{W_{i}}\cap
Z,\overline{W_{i}},i=1,...,r)\\}$ be a collection of algebraic subvarieties of
$X_{2}.$ There exists a Whitney stratification ${\mathcal{S}}$ of $X_{2}$
which is compatible with $\mathcal{R}.$

For any smooth strata $S_{i}\in\mathcal{S}$, let $B_{i}$ be the set of
critical values of the mapping $\overline{f}_{|S_{i}}$ and put
$B=\overline{\bigcup B_{i}}.$ Take $X_{3}=X_{2}\setminus\overline{f}^{-1}(B).$
The restriction of the stratification $\mathcal{S}$ to $X_{3}$ provides a
Whitney stratification which is compatible with the family
${\mathcal{R}}^{\prime}:={\mathcal{R}}\cap X_{3}.$ We have a proper mapping
$f^{\prime}:=\overline{f}_{|X_{3}}:X_{3}\to{Y}\setminus B$ which is a
submersion on each stratum. By Thom first isotopy theorem, there is a
trivialization of $f^{\prime}$ which preserves the strata. It is an easy
observation that this trivialization gives a trivialization of the mapping
$f:X\setminus f^{-1}(B)\to Y\setminus B:=U.$ In particular the fibers
$f^{-1}(y_{1})$ and $f^{-1}(y_{2})$ are homeomorphic via a stratum preserving
homeomorphism. This means that the tuples $(f^{-1}(y_{1}),W_{i}\cap
f^{-1}(y_{1});i=1,...,r)$ and $(f^{-1}(y_{2}),W_{i}\cap
f^{-1}(y_{2}),i=1,...,r)$ are homeomorphic. ∎

######  Corollary 2.6.

Let $X,Y$ be complex irreducible algebraic varieties and $f:X\to Y$ a
polynomial mapping. Let $Z\subset X$ be a constructible subset of $X.$ Then,
there exists a nonempty Zariski open subset $U\subset Y$ such that, for every
$y_{1},y_{2}\in U$, the pairs $(f^{-1}(y_{1}),Z\cap f^{-1}(y_{1}))$ and
$(f^{-1}(y_{2}),Z\cap f^{-1}(y_{2}))$ are homeomorphic.

###### Proof.

Since $Z$ is constructible we have $Z=\bigcup^{s}_{i=1}P_{i}\setminus Q_{i}$,
where $P_{i},Q_{i},\ i=1,...,s$ are Zariski closed. Now we can apply Theorem
2.5, where the family $\\{W_{i}\\}$ coincide with $\\{P_{i},Q_{i};\
i=1,...,s\\}$. ∎

######  Theorem 2.7.

Let $X,Y$ be affine complex varieties and let $f:X\to Y$ be a polynomial
mapping. Then, for every $k\in\hbox{\tenmsy N}$, the set $A_{k}:=\\{y\in
Y:\chi(f^{-1}(y))=k\\}$ is a constructible set.

###### Proof.

We proceed by induction on $d=\dim Y.$ If $d=0$, the result is obvious. Assume
$d>0.$ Let $Y_{1},...,Y_{s}$ be all $d-$dimensional components of $Y$ and let
$Z_{i}=Y_{i}\setminus(\overline{Y\setminus Y_{i}}).$ By Theorem 2.5 there
exists a nonempty open subset $U_{i}\subset Z_{i}$ such that $\chi(f^{-1}(y))$
is constant for $y\in U_{i}.$ Take
$Y^{\prime}:=Y\setminus\bigcup^{s}_{i=1}U_{i}$ and
$X^{\prime}:=f^{-1}(Y^{\prime}).$ By the induction hypothesis, the set
$A^{\prime}_{k}:=\\{y\in Y^{\prime}:\chi(f^{-1}(y))=k\\}$ is constructible.
Let $U_{i_{1}},...,U_{i_{r}}$ be all subset among $U_{1},...,U_{s}$ such that
the fiber of $f$ over $U_{i}$ has Euler characteristic $k.$ Then
$A_{k}=A^{\prime}_{k}\cup\bigcup^{r}_{k=1}U_{i_{k}}$ is a constructible subset
of $Y.$ ∎

###  2.2. Proof of Theorems 1.4 and 1.5

The next technical lemma is motivated by [16, Theorem 2.6]. See Definition 1.2
for the notion of general position.

######  Lemma 2.8.

Let $X^{n},Y^{n}\subset\mathbb{C}^{2n-1}$ be strong complete intersections in
general position. Then, there exists a linear mapping
$L\colon\mathbb{C}^{2n-1}\to\mathbb{C}$ such that

$(\Phi,L):X\times
Y\to(\mathbb{C}^{2n-1}\times\mathbb{C}),(\Phi,L)(x,y)=(\Phi(x,y),L(x))$

is a proper mapping.

###### Proof.

By assumption, $\mathcal{Z}_{X\cap
Y}^{\infty}:=\left(\bigcap_{1}^{n-1}\\{(f_{i})_{*}=0\\}\right)\bigcap\left(\bigcap_{1}^{n-1}\\{(g_{i})_{*}=0\\}\right)\subset\mathbb{C}^{2n-1}$
is a union of finitely many complex lines. Let
$L\colon\mathbb{C}^{2n-1}\to\mathbb{C}$ be a linear mapping such that
$L(z)\neq 0,\forall z\in(\mathcal{Z}_{X\cap Y}^{\infty}\setminus\\{0\\})$. We
claim that

$(\Phi,L):X\times Y\to(\mathbb{C}^{2n-1}\times\mathbb{C}),$

$(\Phi,L)(x,y)=(\Phi(x,y),L(x))$ is a proper mapping.

In fact, let $x(t)=(x_{1}(t),\ldots,x_{2n-1}(t))$ and
$y(t)=(y_{1}(t),\ldots,y_{2n-1}(t))$ be analytic curves such that
$(x(t),y(t))\in X\times Y$, $\Phi(x(t),y(t))\to p\in\mathbb{C}^{2n-1}$ and
$\|(x(t),y(t))\|\to\infty,$ as $t\to\infty$. It follows that
$\|x(t)\|\to\infty$ and $\|y(t)\|\to\infty$, as $t\to\infty$.

We set

$x_{i}(t)=a_{\alpha_{i}}t^{\alpha_{i}}+\mbox{ lower terms,
}y_{i}(t)=b_{\beta_{i}}t^{\beta_{i}}+\mbox{ lower terms},i=1,\ldots,2n-1,$

$\alpha:=\max_{i}\\{\alpha_{i}\\}$, $\beta:=\max_{i}\\{\beta_{i}\\}$,
$a^{\prime}:=(a_{1}^{\prime},\ldots,a_{2n-1}^{\prime})$ and
$b^{\prime}:=(b_{1}^{\prime},\ldots,b_{2n-1}^{\prime}),$ where
$a_{i}^{\prime}=a_{\alpha_{i}}$ (resp., $b_{i}^{\prime}=b_{\beta_{i}}$) if
$\alpha=\alpha_{i}$ (resp., $\beta=\beta_{i}$) and $a^{\prime}_{i}=0$ (resp.,
$b^{\prime}_{i}=0$) if $\alpha>\alpha_{i}$ (resp., $\beta>\beta_{i}$).

Since $\Phi(x(t),y(t))\to p\in\mathbb{C}^{2n-1}$, we have $\alpha=\beta$ and
$a^{\prime}=-b^{\prime}$. From the expansions:

$f_{j}(x(t))=(f_{j})_{\nu_{j}}(a^{\prime})t^{\alpha\nu_{j}}+\mbox{ lower
terms,}$ $g_{j}(y(t))=(g_{j})_{\mu_{j}}(b^{\prime})t^{\beta\mu_{j}}+\mbox{
lower terms},j=1,\ldots,n-1,$

and since $f_{j}(x(t))\equiv 0,g_{j}(y(t))\equiv 0$ and $\mathcal{Z}_{X\cap
Y}^{\infty}$ is a homogeneous set, we have
$a^{\prime},b^{\prime}\in\mathcal{Z}_{X\cap Y}^{\infty}$.

On the other hand, we have the following expansion for $L$:

$L(x(t))=L(a^{\prime})t^{\alpha}+\mbox{ lower terms}.$

Since $a^{\prime}\in\mathcal{Z}_{X\cap Y}^{\infty},$ we have
$L(a^{\prime})\neq 0$ and therefore $\|L(x(t))\|\to\infty$. This shows that
$(\Phi,L)$ is a proper mapping, as desired. ∎

Directly from the proof of Lemma 2.8, we have that if
$L\colon\mathbb{C}^{2n-1}\to\mathbb{C}$ is a linear mapping such that
$L(z)\neq 0$ for any $z\in(\mathcal{Z}_{X\cap Y}^{\infty}\setminus\\{0\\})$,
then the mapping $(\Phi,L):X\times
Y\to(\mathbb{C}^{2n-1}\times\mathbb{C}),(\Phi,L)(x,y)=(\Phi(x,y),L(x))$ is a
proper mapping.

Now, we prove Theorem 1.4.

###### Proof of Theorem 1.4.

The proof is motivated by arguments from [15, Theorem 3.1] and [16, Theorem
2.1].

From Lemma 2.8, there is a linear mapping
$L\colon\mathbb{C}^{2n-1}\to\mathbb{C}$ such that

$(\Phi,L):X\times
Y\to(\mathbb{C}^{2n-1}\times\mathbb{C}),(\Phi,L)(x,y)=(\Phi(x,y),L(x))$

is a proper mapping.

Let $p\notin\overline{K_{0}(\Phi)}$ and consider the smooth algebraic set
$\Phi^{-1}(p)$. The restriction $L_{|\Phi^{-1}(p)}$ is a proper mapping, since
$(\Phi,L)$ is a proper mapping and $\Phi^{-1}(p)$ is a closed set. It follows
that $L_{|\Phi^{-1}(p)}(\Phi^{-1}(p))$ is a closed set and, since it is also a
constructible set, we have that $L_{|\Phi^{-1}(p)}(\Phi^{-1}(p))$ is an
algebraic set (constructible closed sets are algebraic sets). Thus
$L_{|\Phi^{-1}(p)}(\Phi^{-1}(p))=\mathbb{C}$.

We have $S_{(\Phi,L)}=\emptyset$, where $S_{(\Phi,L)}$ denotes the non-
properness set of $(\Phi,L)$. It follows from Theorem 2.4 that
$B(\Phi,L)=K_{0}(\Phi,L)$ and, consequently, $(\Phi,L)$ is a ramified mapping
of degree $d:=\mu(\Phi,L)$, where $\mu(\Phi,L)$ denotes the geometric degree
of $(\Phi,L)$. Thus, for any $p\notin\overline{K_{0}(\Phi)}$ and any regular
value $c\in\mathbb{C}$ of the mapping $L_{|\Phi^{-1}(p)},$ we have
$\\#(L_{|\Phi^{-1}(p)})^{-1}(c)=\\#\\{\Phi^{-1}(p)\cap
L^{-1}(z)\\}=\mu(\Phi,L)=d$.

Take $p_{0}\notin\overline{K_{0}(\Phi)}$ and consider a small enough
neighborhood $W$ of $p_{0}$ such that $W\cap\overline{K_{0}(\Phi)}=\emptyset$.
For any $p\in W$, we set

$\Sigma_{p}:=\\{z_{1}(p),\ldots,z_{r(p)}(p)\\}\subset X\times Y$

and

$\Delta_{p}:=\\{c_{1}(p),\ldots,c_{s(p)}(p)\\}\subset\mathbb{C},$

the critical points set and critical values set of $L_{|\Phi^{-1}(p)}$
respectively.

As in the proofs of Theorem 3.1 of [15], Theorem 2.1 of [16], or from [14,
(III,32)], we obtain the following like Riemann-Hurwitz formula:

(2.1)
$\chi(\Phi^{-1}(p))=d+r(p)-\sum_{i=1}^{r(p)}\rho_{i}(p)=d-\sum_{i=1}^{r(p)}(\rho_{i}(p)-1),$

where $\rho_{i}(p)$ is the multiplicity of the point $x_{i}(p)$. These
equalities imply that there is a small enough neighborhood of $p_{0}$ such
that the following inequality holds

$\chi(\Phi^{-1}(p_{0}))\geq\chi(\Phi^{-1}(p)).$

Therefore, there is a sufficiently small neighborhood $V:=B(p_{0},\delta)$
such that $V\subset W$ and $\chi(\Phi^{-1}(p_{0}))=\chi(\Phi^{-1}(p))$ for any
$p\in V$ if and only if there is no point $x_{i}(p)$ that converges to
infinity as $p$ converges to $p_{0}$.

By similar arguments of [15, Theorem 3.1], [16, Theorem 2.1] and [12, Theorem
3.1], we will show that $\chi(\Phi^{-1}(p_{0}))=\chi(\Phi^{-1}(p))\forall p\in
V$ implies $p_{0}\notin B_{\infty}(X,Y)$.

As we have said, $\chi(\Phi^{-1}(p_{0}))=\chi(\Phi^{-1}(p))\forall p\in V$
implies that $\cup_{p\in V}\Sigma_{p}$ is bounded and, since $(\Phi,L)$ is a
proper mapping, that there exists $r\in\mathbb{R}$ such that
$K:=\Phi^{-1}(V)\cap L^{-1}(\overline{B(0,r)})$ is a bounded set containing
$\cup_{p\in V}\Sigma_{p}$. Then, we set:

$U_{1}=\Phi^{-1}(V)\setminus\overline{K},$

and we will show that $\Phi$ restricted to $U_{1}$ is a fibration on $V$.

Consider the (formal) system of differential equations:

$\left(\begin{array}[]{cc}[df(x(t))]_{(n-1)\times(2n-1)}&[0]_{(n-1)\times(2n-1)}\\\
\\\ {[0]}_{(n-1)\times(2n-1)}&[dg(y(t))]_{(n-1)\times(2n-1)}\\\ \\\
&{[d\Phi(x(t),y(t))]_{(2n-2)\times(4n-2)}}\\\ \\\
&[dL(x(t),y(t))]_{1\times(4n-2)}\end{array}\right)\left(\begin{array}[]{c}x^{\prime}_{1}(t)\\\
\vdots\\\ x^{\prime}_{2n-1}(t)\\\ \\\ y^{\prime}_{1}(t)\\\ \vdots\\\
y^{\prime}_{2n-1}(t)\\\
\end{array}\right)=\left(\begin{array}[]{l}[0]_{(n-1)\times 1}\\\ \\\
{[0]}_{(n-1)\times 1}\\\ \\\ {[p-p_{0}]}_{(2n-1)\times 1}\\\ \\\
{[0]}_{1\times 1}\end{array}\right).$

We can solve the system, since the matrix on left size is invertible (at least
in $U_{1}$) and we can write this system shortly as follows:

$z^{\prime}(t)=V_{1}(p-p_{0},z(t)),$

with $z(t)=(x(t),y(t))$. From above constructions, it follows that:

$df(x(t))x^{\prime}(t)=0,\hskip 5.69046ptdg(y(t))y^{\prime}(t)=0,$
$d\Phi(V(p-p_{0},z(t)))=p-p_{0},\hskip 5.69046ptdL(V(p-p_{0},z(t)))=0.$

Let us consider the following differential equation:

(2.2)
$z^{\prime}(t)=V_{1}(p-p_{0},z(t)),\,\,z(0)=\gamma\in\Phi^{-1}(p_{0})\setminus\overline{B(0,R)}.$

It follows that if $z_{p-p_{0},\gamma}(t)=(x(t),y(t))$ is a solution of (2.2)
then $z_{p-p_{0},\gamma}(t)\in X\times Y$, $L(z_{p-p_{0},\gamma}(t))$ is
constant and $\Phi(z_{p-p_{0},\gamma}(t))=t(p-p_{0})+p_{0}$. Then, from
definition of $U_{1}$ and since $(\Phi,L)$ is a proper mapping, the trajectory
of $z_{p-p_{0},\gamma}$ is bounded and does not cross the boundary of $U_{1}$
for any $t\in[0,1]$. It follows that the trajectory $z_{p-p_{0},\gamma}(t)$ is
defined on whole $[0,1]$ and it is contained in $X\times Y$.

The phase flow $z_{p-p_{0},\gamma}(t)$, $t\in[0,1]$, transforms
$\Phi^{-1}(p_{0})\cap U_{1}$ into $\Phi^{-1}(t(p-p_{0})+p_{0})\cap U_{1}$ (in
fact, by symmetry, it transforms $\Phi^{-1}(p_{0})\cap U_{1}$ onto
$\Phi^{-1}(t(p-p_{0})+p_{0})\cap U_{1}$). The mapping

$h:\Phi^{-1}(p_{0})\cap U_{1}\times V\ni(\gamma,p)\to
z_{p-p_{0},\gamma}(1)\in\Phi^{-1}(V)\cap U_{1},$

is a diffeomorphism. Thus $p_{0}\notin B_{\infty}(X,Y).$

Let $k$ be the Euler characteristic of a typical fiber of $\Phi.$ Hence a
point $p\in\mathbb{C}^{2n-1}\setminus\overline{K_{0}(\Phi)}$ is typical if and
only if $\chi(\Phi^{-1}(p))=k.$ By Theorem 2.7, the set $A_{k}$ of such points
is constructible in $\mathbb{C}^{2n-1}\setminus\overline{K_{0}(\Phi)}.$ Since
$A_{k}$ is open, it is Zariski open. Hence,
$B(X,Y)=(\mathbb{C}^{2n-1}\setminus A_{k})\cup\overline{K_{0}(\Phi)}$ is a
Zariski closed set, i.e., $B(X,Y)$ is an algebraic set. ∎

######  Definition 2.9.

We set

$L_{\infty}(\Phi):=\\{y\in
Y\mid\exists\\{x_{k}\\}\subset\mathrm{Sing}(\Phi,L)\setminus\mathrm{Sing}(\Phi)\
\mbox{ s.t. }\|x_{k}\|\to\infty\mbox{ and }\Phi(x_{k})\to y\\}.$

Let $S:=\overline{\mathrm{Sing}(\Phi,L)\setminus\mathrm{Sing}(\Phi)}$. We have
that $L_{\infty}(\Phi)$ is the non-properness set of the restriction
$\Phi_{|S}:S\to\mathbb{C}^{2n-1}$ and as a direct consequence of the proof of
Theorem 1.4 (after equation 2.1), it follows that:

(2.3) $B_{\infty}(X,Y)\subset L_{\infty}(\Phi).$

Theorem 1.5 follows from the next result:

######  Theorem 2.10.

Let $L$ be a generic linear mapping. Then $L_{\infty}(\Phi)$ is either empty
or a hypersurface such that:

(2.4) $\deg L_{\infty}(\Phi)\leq D+d-\mu_{X,Y},$

where $D:=\deg\overline{\mathrm{Sing}(\Phi,L)\setminus\mathrm{Sing}(\Phi)}$,
$d:=\mu(\Phi,L)$, where $\mu(\Phi,L)$ denotes the geometric degree of
$(\Phi,L)$.

###### Proof.

Consider $S:=\overline{\mathrm{Sing}(\Phi,L)\setminus\mathrm{Sing}(\Phi)}$. If
$S=\emptyset$, then $L_{\infty}(\Phi)=\emptyset$. Suppose $S\neq\emptyset$. We
have $\dim S=2n-1$. Consider the restriction of $\Phi$ to $S$, which is an
equidimensional mapping:

$\Phi_{|S}:S\to\mathbb{C}^{2n-1}.$

Note that for $p\in\mathbb{C}^{2n-1}\setminus K_{0}(\Phi)$ the fiber
$\Phi^{-1}(p)$ is smooth (hence reduced). Hence, if $Q$ is an irreducible
(one-dimensional) component of this fiber, then either the mapping
$L_{|Q}:Q\to\mathbb{C}$ is an isomorphism or it has a finite number of
critical points. In the first case, $Q$ is not contained in $S$. In the second
case, $Q$ has only a finite number of points in common with $S$ (all critical
points of $L_{|Q}$ are in S). This means that the mapping $\Phi_{|}S$ is
generically finite and so dominant.

Let $D:=\deg S$. We can apply Theorem 2.2. Note that the algebraic degree of
$\Phi$ is $1$. Thus,

$\deg L_{\infty}(\Phi)\leq D-\mu(\Phi_{|S}).$

In particular,

$\deg L_{\infty}(\Phi)\leq D-1$

and, by [9, Proposition 2.8], we have $\deg
L_{\infty}(\Phi)\leq\prod^{n-1}_{i=1}a_{i}\prod^{n-1}_{i=1}b_{i}(\sum^{n-1}_{i=1}a_{i}+b_{i}-2)-1.$

By (2.1), we have

$-\mu(\Phi_{|S})=-r(p)\leq-\chi(\Phi^{-1}(p_{0}))+d$

and, consequently, we can obtain the estimation:

$\deg L_{\infty}(\Phi)\leq D-\mu(\Phi_{|S})\leq D-\chi(\Phi^{-1}(p_{0}))+d,$

as desired.

∎

###  2.3. Proof of Theorem 1.6

Denote by $\mathcal{L}(\mathbb{C}^{2n-1},\mathbb{C}^{2n-1})$ the set of linear
mappings from $\mathbb{C}^{2n-1}$ to $\mathbb{C}^{2n-1}$. Let us consider the
mapping $\Psi:X\times
Y\times\mathcal{L}(\mathbb{C}^{2n-1},\mathbb{C}^{2n-1})\ni(x,y,H)\mapsto(\frac{x+H(y)}{2},H)\in\mathbb{C}^{2n-1}\times\mathcal{L}(\mathbb{C}^{2n-1},\mathbb{C}^{2n-1}).$
The mapping $\Psi$ is a polynomial mapping, hence it has homeomorphic fibers
outside the bifurcation set $B(\Psi).$ The Zariski closure
$B:=\overline{B(\Psi)}$ is a proper algebraic subset of
$\mathbb{C}^{2n-1}\times\mathcal{L}(\mathbb{C}^{2n-1},\mathbb{C}^{2n-1}).$

Put
$R=\\{H\in\mathcal{L}(\mathbb{C}^{2n-1},\mathbb{C}^{2n-1}):\mathbb{C}^{2n-1}\times\\{H\\}\subset
B\\}.$ It is easy to see that the set $R$ is algebraic. Since the set $B$ is a
proper algebraic subset of
$\mathbb{C}^{2n-1}\times\mathcal{L}(\mathbb{C}^{2n-1},\mathbb{C}^{2n-1})$, the
set $R$ is a proper algebraic subset of
$\mathcal{L}(\mathbb{C}^{2n-1},\mathbb{C}^{2n-1}).$ Outside $R$ a generic
fiber of the mapping $X\times
Y\ni(x,y)\mapsto(\frac{x+H(y)}{2})\in\mathbb{C}^{2n-1}$ is homeomorphic to the
generic fiber of $\Psi.$ We can finish the first part of theorem from
Proposition 1.3.

Now, we show that

$(\mathbb{C}^{2n-1},B(X,H(Y)))\ {\rm and}\
(\mathbb{C}^{2n-1},B(X,H^{\prime}(Y)))$

are homeomorphic for generic
$H,H^{\prime}\in\mathcal{L}(\mathbb{C}^{2n-1},\mathbb{C}^{2n-1}).$

Denote $\Psi_{H}(x,y):=\Psi(x,y,H).$Given a mapping $f$, let $C(f)$ denote the
set of critical points of $f.$ It is easy to compute the
$C(\Psi)=\bigcup_{H\in\mathcal{L}(\mathbb{C}^{2n-1},\mathbb{C}^{2n-1})}C(\Psi_{H})\times\\{H\\}.$

In particular, if
$\pi:\mathbb{C}^{2n-1}\times\mathcal{L}(\mathbb{C}^{2n-1},\mathbb{C}^{2n-1})\to\mathcal{L}(\mathbb{C}^{2n-1},\mathbb{C}^{2n-1})$
is a projection, then $\pi^{-1}(H)\cap K_{0}(\Psi)=K_{0}(\Psi_{H}).$ Since
$K_{0}(\Psi)$ is a constructible set, it follows from Theorem 2.5 (or rather
from their proof) that
$\pi^{-1}(H)\cap\overline{K_{0}(\Psi)}=\overline{K_{0}(\Psi_{H})}.$

Let $k$ be the Euler characteristic of a typical fiber of $\Psi$ and let
$B_{k}=\\{y\in\mathbb{C}^{2n-1}\times\mathcal{L}(\mathbb{C}^{2n-1},\mathbb{C}^{2n-1}):\chi(\Psi^{-1}(y))\not=k\\}.$
By Theorem 2.7, the set $B_{k}$ is constructible. Moreover,
$B(\Psi_{H})=\pi^{-1}(H)\cap(B_{k}\cup\overline{K_{0}(\Psi)})=B(X,H(Y)).$ Now,
the result follows from Corollary 2.6.

## Acknowledgments

The first author was partially supported by the Fapemig-Brazil Grant
APQ-02085-21 and CNPq-Brazil grant 301631/2022-0. The second author was
partially supported by the grant of Narodowe Centrum Nauki number
2019/33/B/ST1/00755. The authors are grateful to professor M.A.S Ruas for many
helpful discussions.

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