Learn Hardcore Hungarian: A jegyem az asztalon van.

Hungarian normally uses the definite article with possessed nouns: A jegyem = “the my-ticket.” This is standard and sounds natural. You don’t usually drop the article here. You would drop the article in certain existential/possessive constructions like Van jegyem (“I have a ticket”), but not in a simple statement like this.

Why is it a in A jegyem but az in az asztalon?

Hungarian has two forms of the definite article:

a before consonant-initial words (e.g., a jegyem)
az before vowel-initial words (e.g., az asztalon, because asztal starts with a vowel sound)

Both mean the.

Does az here mean “that,” as in “that table”?

No. In az asztalon, az is just the definite article (the). If you want “that table,” use a demonstrative: azon az asztalon (“on that table”), where azon = “on that (one)” and az asztalon = “on the table.”

Can I omit van the way Hungarian often omits “to be” in the present?

Not here. In Hungarian, the copula is usually omitted in 3rd person present with noun/adjective predicates (e.g., Ő tanár = “He is a teacher”). But with locations or existence you must use van/vannak: A jegyem az asztalon van is required.

What are acceptable word orders, and what do they emphasize?

A jegyem az asztalon van. Neutral/typical: topic (A jegyem) + focus (az asztalon) + van.
Az asztalon van a jegyem. Focus on location (“It’s on the table that my ticket is”).
A jegyem van az asztalon. Focus on subject (“It’s my ticket that is on the table, not something else”).

Hungarian uses word order to mark focus; the focused element sits right before the verb.

Why is it asztalon and not asztalra or asztalnál?

asztalon (superessive -on/-en/-ön): on top of something, static location.
asztalra (sublative -ra/-re): onto something, motion toward a surface. Example: A jegyet az asztalra teszem (“I’m putting the ticket onto the table”).
asztalnál (adessive -nál/-nél): at/by/near something. Example: Az asztalnál ülök (“I’m sitting at the table”).

How do I negate this sentence?

Two common ways:

Neutral negation of presence: A jegyem nincs az asztalon. (“My ticket is not on the table.”) Use nincs (not “nem van”).
Contrastive negation (focusing the location): Nem az asztalon van a jegyem, (hanem …) (“It’s not on the table, but …”). Here nem negates the focused phrase.

How do I make it plural (“My tickets are on the table”)?

A jegyeim az asztalon vannak. Notes:
jegy → jegyeim (“my tickets”: plural + 1sg possessor)
van → vannak for a plural subject

How do I ask “Where is my ticket?” or make a yes–no question?

“Where is my ticket?” → Hol van a jegyem?
Yes–no: A jegyem az asztalon van? (intonation does the work). You can also say Van a jegyem az asztalon?, but the first is more neutral here.

How do I say “on a table” instead of “on the table”?

Use the indefinite article: egy asztalon. Examples:

A jegyem egy asztalon van. (“My ticket is on a table.”)
Egy asztalon van a jegyem. (focus on “a table”)

Why is the vowel in jegyem an e? Could it be jegyom/jegyöm?

Hungarian vowel harmony determines the linking vowel before the 1sg possessive -m:

Front-vowel words take -em/-öm.
Back-vowel words take -om. Since jegy has front vowels, you get jegyem. Compare:
házam (“my house”: back vowel)
székem (“my chair”: front unrounded)
könyvem (“my book”: front rounded)

Any pronunciation tips for this sentence?

Word stress is always on the first syllable: A JEGYem az ASZtalon van.
gy in jegyem is a soft “dy” sound.
sz is like English “s”; Hungarian s (not present here) is “sh.”
az asztalon: the two adjacent a’s are pronounced separately; say it smoothly like “az asz-.”

Does jegy ever mean something other than “ticket”?

Yes. jegy can also mean a school grade/mark, or a sign/mark in other contexts. In everyday contexts with asztal, it’s understood as “ticket.” Context disambiguates.