Johan van Waveren Hudde (1628 - 1704); Dutch Mathematician - born in Amsterdam.

Hudde studied the art of mathematics at the University of Leiden (oldest university of the Netherlands, founded 1575 by Willen van Oranje. Willem was a leader of the resistance against the Spanish influence in the Netherlands. He was killed by a catholic fanatic in 1584). Hudde made very important contributions in many areas like polynomial roots, minima and maxima etc. His manuscripts had a crucial influence on Leibniz's introduction to calculus. Hudde was the first who introduced an algebraic solution for the

Any solution x we can write as x = 0.5*(k + h + m). k,h,m some complex numbers

In other words:

4x

16x

Using the free dimensions of x we set: (i) k

2x

2x

2x

/P normed polynomial x^{4} + ax^{3} + ...

`a`b`c`d set' 0.25 1 1 1f*1_P

`p`q`r set'(b-6f*l),(c-2*a*b-4*l),d+a*(neg c)+a*b-3*l:a*a

/solving cubic resolvent

.cub.s:1f,(2*p),((p*p)-4*r),neg q*q

/all solutions of the resolvent in the complex plane

t:.cub.r

/find a square root of t

tx:.complex.arootz (flip t;2)

/condition (ii)

if[(signum first .complex.prdz tx)=signum q;tx:(1 1 -1)*/:tx]

/final shift and solution

sol:((neg a),0f)+0.5*flip sum flip M:(flip tx)*/:(1 1 1;1 -1 -1;-1 1 -1;-1 -1 1)

`a`b`c`d set' 0.25 1 1 1f*1_P

`p`q`r set'(b-6f*l),(c-2*a*b-4*l),d+a*(neg c)+a*b-3*l:a*a

/solving cubic resolvent

.cub.s:1f,(2*p),((p*p)-4*r),neg q*q

/all solutions of the resolvent in the complex plane

t:.cub.r

/find a square root of t

tx:.complex.arootz (flip t;2)

/condition (ii)

if[(signum first .complex.prdz tx)=signum q;tx:(1 1 -1)*/:tx]

/final shift and solution

sol:((neg a),0f)+0.5*flip sum flip M:(flip tx)*/:(1 1 1;1 -1 -1;-1 1 -1;-1 -1 1)

Example: x

So:

P:P%first P:1 -10 34 54 -495

⇒ p = -3.5 ; q = 99 ; r = -264.6875⇒ resolvent: t

⇒ solutions: x

Lodovico Ferrari (1522 - 1565); Italian Mathematician - born in Bologna.

Lodovico was an orphan since early childhood and his uncle Vincenzo took care of Lodovico; but facts are that Vincenzo looked at Lodovico as kind of equity only, and in order to get rid of him - he sent Lodovico (14 years old) to Cardano as a servant. Cardano very soon recognized Lodovico's very keen mind, but in addition Lodovico could read and write. Cardano soon freed him from all servant duties and introduced Lodovico in Latin, Greek and mathematics.

By adding to both sides 0.25a

Example: 15x

/eliminate numerical noise

flatn:{@[x;where 1e-12>abs x;:;0f]}

/Polynomial P:P%first P:15 -82 4858 -26650 -5525

`a`b`c`d set' 1_P

.cub.s:1f,(neg b),((a*c)-4*d),(4*b*d)-(c*c)+d*a*a

/back to Renaissance

\l cardanoTartagliaFerro.q

/solutions t

t:.cub.r

/choosing a real solution ..

zeta:first first t

/Left side of the square

L:1f,0.5*a,zeta

/addressing the context of quadratic equations

w:`.quadr.m`.quadr.n`.quadr.q

w set' flatn ((neg b)+zeta+0.25*a*a;(neg c)+0.5*a*zeta;(neg d)+0.25*zeta*zeta)

/quadr.equation result complex-valued r=(RE,IM)

\l quadr.q

/full square handling, i.e. if a = c = 0

cofactor:1f;factor:.quadr.m;if[1e-13>factor;factor:1f;cofactor:0f]

/"sqrt factor" undoing the normalization

l:(sqrt factor)*0f,cofactor,neg first first .quadr.r (U,V values)

/solutions 1 and 2

w set' L+l

\l quadr.q

l12:.quadr.r

/solutions 3 and 4

w set' L-l

\l quadr.q

l34:.quadr.r

flatn:{@[x;where 1e-12>abs x;:;0f]}

/Polynomial P:P%first P:15 -82 4858 -26650 -5525

`a`b`c`d set' 1_P

.cub.s:1f,(neg b),((a*c)-4*d),(4*b*d)-(c*c)+d*a*a

/back to Renaissance

\l cardanoTartagliaFerro.q

/solutions t

t:.cub.r

/choosing a real solution ..

zeta:first first t

/Left side of the square

L:1f,0.5*a,zeta

/addressing the context of quadratic equations

w:`.quadr.m`.quadr.n`.quadr.q

w set' flatn ((neg b)+zeta+0.25*a*a;(neg c)+0.5*a*zeta;(neg d)+0.25*zeta*zeta)

/quadr.equation result complex-valued r=(RE,IM)

\l quadr.q

/full square handling, i.e. if a = c = 0

cofactor:1f;factor:.quadr.m;if[1e-13>factor;factor:1f;cofactor:0f]

/"sqrt factor" undoing the normalization

l:(sqrt factor)*0f,cofactor,neg first first .quadr.r (U,V values)

/solutions 1 and 2

w set' L+l

\l quadr.q

l12:.quadr.r

/solutions 3 and 4

w set' L-l

\l quadr.q

l34:.quadr.r

l12 ... ± 18.02776i (= ± 5i*13

l34 ... 0.2 and 5.66666...

Other Examples ...

Sample 1: x

Sample 2: x

Sample 3: x

Sample 4: x

Sample 5: x

Joseph Louis Lagrange (1736 - 1813); Italian/French Mathematician - born in Turin, died in Paris.

Lagrange was born as the oldest of 11 children (of which only 2 survived to adulthood). Lagrange (he was an autodidact, and did not have the opportunity to study with world-class mathematicians) is without any doubt one of the most gifted mathematicians of the 18th century. With the age of 16 he became professor of mathematics at the

Let be:(+++)

a

a

a

etc..

a

For any symmetric polynomial Θ(x

So let be now σ

ξ

ξ

ξ

Important to know at this point: If we permute the σ

The expressions (ξ

ξ

ξ

ξ

ξ

ξ

This means ξ

and consequently we end up with:

σ

σ

We then continue with ξ

©++ MILAN ONDRUS