Lecture 21

Today we will talk about numerical functions and polynomial like

functions .

Def " : A numerical function is a function f : I→ Q .

Difference operator : If G =L numerical functions} , the difference

operator I :b → 6

is the functionf ↳ If where If (n) E f (nth - f- Cn) .

Example : The binomial polynomials Q pdx) , 12=0,1 , . . .

,are :

Qocx) = I

Q , ( x) =X .

Qa Cx) = X (x-D

-2.1"

O

O

O

QIN = XCx-DktDk !

O

:

define numerical functions It → Q that are infract integer -valued .

Note Qnlx) has degree K and leading coefficient 11kt.

. Consequently ,

Exercise : { QIN : k ETI>o} is a Q - basis of QEII .

Lemma l : Considering Qu (x) as a numerical function I→ Q ,

we have

AQ,

= Qe ..

Pf : Omitted .

I

Def": A numerical function f : I→ Q is polynomial like

or of polynomial type if 7 a polynomial Pf (x) C- ④ Ex]

sit . It n 770 ,

f- ( n ) = Pf Cn) .

Note Pg (x) is uniquely determined by f because ifP, , Pz E QQ] agree for infinitely many values of X ,

then 13=12 .

We say Pfk) is associated to f and define degree of f , denoted degf , to be

the degree of Pf Cx) .

Note that if dey f = d ,then 7 ! expression

Pg Cx) =

ad Qdlx) t ad. , Qd. , CN t . . - t ao Qolxl ; a ; EQ

using the fact that the binomial polynomials form a Q - basis of① GI .

The coefficient ad is called the multiplicity of f , and denotedmutt f .

Note if deg f =D ,then mutt f = d ! x leading coefficient of Pfcx) .

Lemma 2 : Let f : I-7 Q be a numerical function .

C) If (n) = O for n>> o ⇒ f is polynomial like withassociated polynomial a constant .

(2) f is polynomial like ⇐ If is polynomial like . In this case

if Pf is the associated polynomial of f ,then Ipf is

the associated polynomial of Af . Furthermore, if deg Af 7,0

then deg f = It deg Af and mutt f = mutt If .

Pf : C ) This is clear.

(2) If f is polynomial like with associated polynomial Pf , thenAf is polynomial like with associated polynomial Ipf .

Conversely , suppose If is polynomial like with associated

polynomialPaf = ad Qdlx) t ad. , Qd. .CH t - . - tao ; ai E Q .

Lemma I

=a,aAQd*LN t ad. ,

A- Qdcx) t . . - ta.AQ

,(x)

= A ( adQa+ , LH t ad. , Qd (N t . .. tao Q ,

CH ) .

Let P 8= ad Qa+ ,(N t ad. , Qd (x) too . t ao Q ,

(x).

Then H noo, Ifk ) = Pain) = AP (n) ⇒ H n ⇒ O

, Iff - p) (n) = O .

⇒ f -P is polynomial like with associated constant polynomial F=b .

Then f is polynomial like with associated polynomial Ptb .

Note that Ptb has degree dog Pdf tl = deg Af tl andsince

Ttb =

ad Qdt ,CN t ad. , Qdlxl t . . - t ao Q ,CN t b

,

mutt f Ead

= mutt Af .

I

Def" 8 A polynomial like function f : I→ Q is

- non - negative if f- (n) 7,0 H n ⇒ 0.

- integer - valued if fcn) EI H n >70 .

Example : The numerical function defined by each binomial polynomialQr (x) is non- negative and integer -valued .

Lemma 3 : Let f : I → Q be a polynomial like function with

associated polynomialPf =

ad Qdlxt t ad. , Qd . . ( x) t - - - tao ; a it Q .

TFAE :

① f is integer -valued .

② Pf (n) E Z H n → 0 .

③ Pf (n) EI H ne Tt .

④ Ao , . . .

, ad E Tt .

Pf : ① ⇒ ② is dear ; ④ ⇒ ① follows because each QKCNis integer - valued .

② ⇒ ④ : Use induction on the degree d of Pf , with d=o

being obvious . Now, if Pf (n) E TL, H n >70

,then

Atf (n) EI , H na> 0 .

But Ipf has degree d- I if d > I since

A- Pf (X1 =

ad Qd. .CH t ad. , Qd. . LN to . - ta,

.

f. By inductive hypothesis , ad , ad. , , . . .

,a

,E I -

Choose no >> O s - t . Pf (no) E E .Then

Ao = Pfcno) - ( ad Qd(no) t . . - ta,Q

,(not) C- It

because QIN is integer - valued for each k.

O

bo Ad , ad. , s. . .

,a, ,ao E Tt , and we win by induction .

Clearly , ④ ⇒ ③ ⇒② . % We are done !

II

Corollary 4 : If f is an integer - valued polynomial like numerical function ,then mutt f E I .

Moreover, if f is non - negative ,

then muttf y, O ,

with mutt f =0 ⇐ fcn) -- O H n xo .

Pf : Exercise .

I

Note mutt f = degf ! x leading coefficient of Pf .

ooo mutt f =lime fln)

-dim flute)

,for a fixed

-

-

-

n-7N hdegf n→a ndegf- -

CEI .

degf ! degf ! **

Using this observation,we get the following result :

Lemma 5 : Leet f , , f , be non-negative numerical functions of polynomialtype . Let a

,b,c,d,e C- I with a

,c > 0 .

Then

C) f , ( an tb) 7, f , Contd) H n no ⇒ deg f , 7, deg fz .

③ f. ( ntb) 3, fz ( ntd ) 7, f. ( rite ) ft n DO ⇒ degf , = degfa

and multf , = mutt fz .

Pf : C) deg f , = deg fz follows by (1) .Then

,

H n >> O, filntbl-y-fzcntdl-zf.tn# .

ndegf , ndegfz

ndegf ,

- - -

deg fi ! degfz ! dcgf . !

o: Taking thin

,

andusing *& gives us

omultf , 7, mutt fz 7, melt f , .

(1) Exercise.

I

Next time : Hilbert polynomial , Samuel polynomial .