1. Eduardo Enrique Escamilla Saldaña
15, 2014
Monterrey, Nuevo Leon, Mexico
February
Purdue Problem of the Week No.4
Let 𝑝 be a polynomial in the variables 𝑥! , 𝑥! , … , 𝑥! . Show that if there is a
number 𝐶 such that 𝑝(𝑥! , 𝑥! , … , 𝑥! ) ≤ 𝐶 for all real 𝑥! , 𝑥! , … , 𝑥! then there is
a number r such that 𝑝(𝑥! , 𝑥! , … , 𝑥! ) = 𝑟 .
Proof:
If there is a number 𝐶 such that 𝑝 𝑥! , 𝑥! , … , 𝑥! ≤ 𝐶 for all real 𝑥! , 𝑥! , … , 𝑥! ,
then ∀𝑖, 𝑝! 𝑥! = 𝑝 1,1, … , 𝑥! , … ,1 ≤ 𝐶
In other words for all 𝑖 ∋ 1 ≤ 𝑥! ≤ 𝑛 we have a single-variable polynomial
𝑝! (𝑥) with a finite degree.
Now,
!"#(!! )
!
𝑝! 𝑥! =
𝛼!" 𝑥!
!!!
where ∀𝑖, 𝑗 ∈ ℤ, 𝛼!" ∈ ℂ, 𝑥! ∈ ℝ
!"#(!! )
!"#(!! )
!
𝑝! 𝑥! =
!
𝛼!" 𝑥! =
!!!
!"#(!! )
!
𝛼!" 𝑥! =
!!!
𝛼!" 𝑥!
!!!
where 𝑝! 𝑥! is the complex conjugate of 𝑝! 𝑥!
𝑝! 𝑥! 𝑝! 𝑥! =
!!"(!! )
!!!
!"#(!! )
!!!
!
𝛼!" 𝑥!
𝛼!" 𝛼!" 𝑥 ! −
𝑑𝑒𝑔(𝑝! )
!!!!!"#(!! )
𝛼!" 𝑥!! =
(𝛼!" 𝑥 ! − 𝛼!" 𝑥 ! )(𝛼!" 𝑥 ! − 𝛼!" 𝑥 ! )
!!!!!!!"#(!! )
=
𝑑𝑒𝑔(𝑝! )
𝛼!"
!!!!!"#(!! )
!
𝑥! −
(𝛼!" 𝑥 ! − 𝛼!" 𝑥 ! )(𝛼!" 𝑥 ! − 𝛼!" 𝑥 ! )
!!!!!!!"#(!! )
=

2. 𝑑𝑒𝑔(𝑝! )
𝛼!"
!!!!!"#(!! )
!
𝑥! −
(𝛼!" 𝑥 ! − 𝛼!" 𝑥 ! )
!
1
!!!!!!!"#(!! )
𝑝! (𝑥! ) !
=
which is a polynomial of real coefficients and degree 2𝑑𝑒𝑔(𝑝! )
Since a real polynomial with degree greater than 1 is not bounded it follows
that 𝑝! (𝑥! ) ! is unbounded if 2𝑑𝑒𝑔(𝑝! ) ≥ 1 or
𝑑𝑒𝑔(𝑝! ) ≥ 1/2
otherwise the condition 𝑝! 𝑥! = 𝑝 1,1, … , 𝑥! , … ,1 ≤ 𝐶
wouldn't hold.
But the polynomial a positive integer number of coefficients therefore
∀1 ≤ 𝑖 ≤ 𝑛, 𝑑𝑒𝑔(𝑝! ) = 0 in other words,
𝑝! 𝑥! = 𝛼!! where 𝛼!! is a constant.
Claim: 𝑝! 𝑥! = 𝑝 𝑥! , 𝑥! , … , 𝑥! = 𝑟
Proof: By induction
𝑝! 𝑥! = 𝛼!! = 𝑝 𝑥! , . . . ,1 = 𝑟!
Suppose
𝑝(𝑥! , 𝑥! , . . . , 𝑥!!! , 1) = 𝑟!!!
then:
𝑝(𝑥! , 𝑥! , . . . , 𝑥! ) = 𝑟!!! 𝑥! + 𝜆
but
𝑝! 𝑥! = 𝑟! = 𝑝(𝑥! , 𝑥! , . . . , 𝑥! ) = 𝑟!!! 𝑥! + 𝜆
implying that
𝑝(𝑥! , 𝑥! , . . . , 𝑥! ) = 𝑟!!! 𝑥! + 𝜆 = 𝑟
1
Graham, Knuth, and Patashnik "Concrete Mathematics (A foundation for computer
science) 2nd edition" page 38.